Math MC Quiz #2

Which of the following choices displays the terms of 36 + x2 - 12x in descending order arrangement?

x2 - 12x + 36

...And finally, a completely factored form (so both binomial factors; the difference and sum) for 4x2 - 9y2 is given by _______ .

(2x - 3y)(2x + 3y)

Which selection elicits the completely (thus, simplified too) factored form for the "Difference of Squares" given by (x + 7)2 - (y - 6)2 ? (HINT: Put the ideas of the previous few questions together to accomplish this.)

(x - y + 13)(x + y + 1)

After first removing the GCF (so you will need to do that), which response would then require further factoring because a "Difference of Squares" is revealed - to your now attuned keen eye - as the expression (collection) of remaining terms?

12x2 - 27y2

Whenever a GCF is either nonexistent or has previously been removed, always consider next whether you need to factor some sort of SPECIAL PRODUCT, one of which is a "Difference of Squares". DO NOT FORGET TO DO SO; train yourself to consciously look for ALL special products! Which of the following is neither, as stated, nor will result in a "Difference of Squares" upon removal of a GCF? (For some responses then, you may need to remove any existing GCFs before deciding.)

12x2 - 6y2

So which of these following algebraic expressions portrays a difference?

3x-5y

Remove each monomial GCF (even if an assumed +1 or -1) from the respective groups of the previous correct answer. Which selection depicts the result of this instruction? (HINT: Because they are monomials, use the "Tri-Check" for each GCF.)

3x2(x + 2) - 4(x + 2)

Which of the following trinomials is NOT given in descending order?

4x2 + 9 - 12x

Arrange the answers of the previous question as steps in the order that makes them MOST efficient in the process of factoring any trinomial. (REHEARSE THIS (correct) ARRANGEMENT AS YOU WORK RELEVANT ASSIGNED PROBLEMS UNTIL IT BECOMES A SUBCONSCIOUS ROUTINE FOR FACTORING TRINOMIALS.)

B, A, D, C

Assume that one factor of a "Difference of Squares" is 7x - 6y. By itself, this binomial happens to express a difference among the terms 7x and 6y. In accordance with the specific relationship that exists among the two factors of a "Difference of Squares", the remaining factor MUST be a sum of these very same two terms. Which response comprehensively portrays the remaining factor? (HINT: Addition is a commutative operation, so equivalently-valued sums are a possibility.)

Either A or C

Of all the individual monomial factors within +2FL (as included among FF + 2FL + LL), which of them is (are) capable of including a NEGATIVE sign, due strictly to the UNALTERABLE numerical factor that is involved?

Either A or C

Which selection will satisfy for the missing +2FL term in order to make 16x2 _____ + 9 a "Perfect Square Trinomial"?

Either B or C

WHENEVER YOU ARE EXPECTED TO FACTOR ANY TRINOMIAL, PROCEED WITH THESE STEPS IN THE ORDER PRESENTED: A-L-W-A-Y-S write the trinomial in descending order, if not already so. Next, remove the GCF if one exists. Then inspect the GCF-less trinomial (so either one that had no GCF to begin with or the remaining trinomial factor with the GCF now removed) to see whether it is a perfect square. Remember, "Perfect Square Trinomials" must fit the pattern TERM-FOR-TERM.

FF + 2FL + LL

For the sake of ensuring equivalency to a given factorable four-term polynomial, it is A-L-W-A-Y-S wise to use only a + sign to "separate" the two groups (quantities) that must be established. Thus for the ensuing step, particularly in regard to which version of a presumed ought to be removed from the second of these groups, both options are in play. It might be a positive , or it might be a negative .

GCF

Which is the lowest-degree term within -24x - 4 + 9x2 ?

It is still -4 because the terms of the trinomial from the previous question are no longer presented in descending order here.

If confined to the Real-number system, is it possible to have a negative number as a perfect square?

No. Upon multiplying any two Real-number (or algebraic representatives of Real-numbers) identical factors together, the result is either a positive number or zero; never a negative number.

Are there any obvious differences among the answer choices given in the last question?

Only that the Associative Property has been applied differently as far as which terms and how many terms have been grouped together.

Whether a removable GCF exists or does not, often there is other or further factoring of a polynomial that is possible. Such deferred or additional factoring depends entirely on recognizing whether that polynomial is one of a few specific factorable types, a.k.a, a special product. One such special product is known as a "Difference of (Two) Squares". Which operation is always suggested by the word "difference", and is therefore worthy of note for purposes of the all-important aspect of recognizing certain special products?

Subtraction

From the perspective of the previous five questions, why isn't x2 - 14x - 49 a "Perfect Square Trinomial"?

The lowest-degree term, -49, is not a perfect square; no two identical factors could result in a product of negative 49.

For the "Perfect Square Trinomial" 4x2 - 20x + 25, assume that F is designated as 2x in order to satisfy for FF being 4x2. However, this now means that L will have to be designated as -5, rather than +5, in order to satisfy for +LL being +25 .

True. Thus, +2FL is satisfied by the result of multiplying +2(2x)(-5) because it exactly gives -20x, which has the proper sign for the middle term of the given trinomial. This would not happen if +5 was designated as L.

REMEMBERING TO ALWAYS CHECK YOUR WORK, how can you be absolutely certain that you have answered the previous question correctly?

Use the FOIL method for multiplying two binomials together (or at least take care to make sure that each term of one binomial distributes to each term of the other binomial). If the resulting simplified product is exactly equivalent to the initial "Difference of Squares", 4x2 - 9y2, then the chosen factorization must have been correct.

So despite +2FL beginning with a POSITIVE 2, is it possible for this entire term (the product of three individual monomials), as incorporated within FF + 2FL + LL, to have an overall NEGATIVE value? For example, could -6xy satisfy for the +2FL term as well as +6xy ?

Yes. According to the rules of signs which govern multiplication: Either F or L (but NOT both) would need to include a negative sign in order for the product +2FL to also include a negative sign.

For the trinomial 16x2 + 40x + 25 is it possible for +40x to satisfy as +2FL ?

Yes. As one of two identical factors, F's value can be correctly deduced as 4x and likewise, L's value as +5. Using these determined values, the term that aligns with +2FL is calculated from +2(4x)(+5). This results in +40x exactly.

For the trinomial 9x2 + 12x + 4, is it possible for +4 to satisfy as LL ?

Yes. In fact, there are two pairs of identical factors that would satisfy for LL; either 2 and 2, or -2 and -2 have the ability to form a +4 product.

Does 64x2 - 48x + 9 align TERM-FOR-TERM with the required FF + 2FL + LL pattern for a "Perfect Square Trinomial"? (HINT: Choose carefully!)

Yes. In this case, when F is 8x, then L must be -3 in order for the product of +2FL to satisfy as -48x .

If a common factor is a binomial, must it be surrounded by parentheses when it is factored out (removed) from several polynomial products ?

Yes. Otherwise, it would appear/mean that only the second of the binomial's two terms had been factored out rather than the entire binomial, thus causing an inequivalency.

Whenever a trinomial is capable of being factored beyond simply the GCF, the "additional" factors obtained WILL A-L-W-A-Y-S BOTH BE TWO-TERM POLYNOMIALS, more specifically known as ?

binomials

Begin factoring 50x2 - 8y2 by removing (as you should) the GCF, which is 2. Having done so, and keeping in mind that further factoring may still be possible, WHAT MUST NOW NOT ESCAPE YOUR ATTENTION about the other factor, the expression (collection) of remaining terms?

it's a "Difference of Squares"

In applying the mnemonic (memory) device known as the FOIL method, the first letter "F" stands of for the the result obtained from multiplying the first term of one binomial by the first term of the other binomial. And, the ending letter "L" stands for the the result obtained from multiplying the term of one binomial by the term of the other binomial.

last

A DEFINITIVE identification of any "Perfect Square Trinomial" requires more than noticing that the highest and lowest-degree terms are themselves. However, it is a good place to start because you can often quickly rule out many trinomials as being perfect squares when this is NOT the case.

perfect squares

Subsequent to considering all other information/evidence mentioned in the previous fourteen questions: By the same meanings designated for "F" and "L" in the FOIL method, another mnemonic device is able to be formed from them that will ultimately determine whether a trinomial qualifies as a perfect .

square

By virtue of the meaning of the prefix "tri", a trinomial is a polynomial consisting of exactly qualifying terms.

three

What is the minimum number of items (among numbers, monomial terms, algebraic quantities, or any combination thereof) necessary to form a single difference?

two

All that remains to accurately establishing each identical binomial factor for 4x2 + 12x + 9 is to check on the correctness of the sign that separates the First and Last terms within each such factor. Because these binomials will be identical to one another, their incorporated signs have got to be identical (to each other) as well. It turns out that these signs will A-L-W-A-Y-S exactly match the sign of the +2FL term of the "Perfect Square Trinomial", which in this case is the + from "+ 12x". Yet, because the sign of the +2FL term is always owing directly to whichever sign you have already determined for L, you can simply "attach" that sign to each L, thus completing the second term within each binomial factor. Now, using all information from the previous two questions as well as this one, which response gives the complete and (FOIL-checked?) equivalent factors for 4x2 + 12x + 9 ?

(2x + 3)(2x + 3)

Based upon the requisite types of identical factors, which response presents a factorization befitting a perfect square trinomial? (FYI: A "Perfect Square Trinomial" is another special product, easily DISTINGUISHABLE FROM a "Difference of Squares" by the number of terms in each.)

(2x - 5y)(2x - 5y)

Which selection gives the completely factored form for 16x8 - 81y8 ? (HINT: Be aware that a re-application of the factoring involved in a "Difference of Squares" may be necessary for any initially obtained factor that is revealed to be a "Difference of Squares" itself.)

(2x2 - 3y2)(2x2 + 3y2)(4x4 + 9y4)

Which selection corroborates 9x2 - 24x + 16 by giving the complete and equivalent factored form of it? (Competency demands that you carefully FOIL to provide evidence of equivalency.)

(3x - 4)(3x - 4)

Which response cannot possibly be a factored form for a "Perfect Square Trinomial"? (HINT: Recall that the Commutative Property holds for Addition, but NOT Subtraction.)

(3x - 4y)(4y - 3x)

Assuming that a GCF has either first been removed or had never existed, both the number and types of factors which are obtained from what is (now) a recognizably qualifying "Difference of Squares" A-L-W-A-Y-S follow the same general pattern: There will be precisely two factors; one of which expresses a difference, the other a sum, of the exact same two terms (or quantities) for both factors. Which response gives such a factored form befitting a "Difference of Squares"?

(3x - 5y)(3x + 5y)

Form an EQUIVALENT expression that uses two groups of two terms each for 3x3 + 6x2 - 4x - 8, such that an obvious (not implied) monomial GCF will exist (ostensibly for later removal) in at least one of these groups. Which response fulfills these conjoined directives? (HINT: Check on the equivalency of your selection by removing parentheses properly (so distributing where necessary, especially with regard to signs) to see whether all terms among 3x3, +6x2, -4x, and -8, despite their order, are restored.)

(3x3 + 6x2) + (-4x - 8)

Where factoring four-term (none of these terms alike) polynomials are concerned, removal of a GCF that exists among all four terms is NOT immediately necessary. In fact, doing so WILL MAKE THE OVERALL FACTORING MORE PRONE TO ERRORS! Instead, the approach to factoring such polynomials will begin with "grouping" the terms. Basically, the Associate Property (of Addition) is utilized to create two groups (or quantities), each consisting of two terms (thus equivalently accounting for all four terms). (FYI: Parentheses are typically used to denote "groups".) Which response displays two groups of two terms each?

(3x3 + 9x2) + (-x - 3)

Regardless of the signs of each term within a four-term polynomial, in this class you will be expected to A-L-W-A-Y-S use a + sign between groups. As a consequence of this, the sign that provided the preliminary detail for each of the initial (not-yet-grouped) four terms will be maintained, (but) WITHIN the groups. Thus, once groups are formed, each term's sign will appear inside one set of parentheses or the other and nowhere else. In doing so for 4x3 + 8x2 - 3x - 6, where the first two terms are to occupy one group and the last two terms occupy the other, the resulting equivalency would be expressed as ?

(4x3 + 8x2) + (-3x - 6)

With this in mind (and hopefully hereafter), check your answer to the previous problem as if it was your first line of work in the process of factoring 4x3 + 8x2 - 3x - 6. Do so by "dropping the parentheses" to see whether the initial four-term polynomial is restored, thus verifying equivalency. Specifically, each group's set of parentheses can only be eliminated by distributing (even if conceding to the assumed values +1 or -1) over them completely. After this close examination of your previous answer, which response is (still?) equivalent, yet also in accordance with having only a + sign between groups?

(4x3 + 8x2) + (-3x - 6)

Which response gives the only possible way of forming two groups of two terms each for x with this further qualification: That there is a GCF among the terms for at least one of these groups? (FYI: This consideration must be appeased before proceeding further with this method of factoring four-term polynomials.)

(ax + 5a) + (by + 2b)

Factor out the binomial GCF from the correct response to the last question. Which choice satisfies this mandate? (HINT: For the sake of equivalency, FOIL your answer to ensure that all terms among 3x3, +6x2, -4x, and -8, regardless of order, are restored.)

(x + 2)(3x2 - 4)

Although the typically encountered "Difference of Squares" assumes the form of a binomial, this is NOT a requirement. The "Squares" component of this special product may individually be ANY perfect square, such as a squared quantity. (NOTE: A mathematical "quantity" will mean a multiple-term expression that is surrounded by either obvious or implied parentheses.) Which of the responses depicts such a squared quantity?

(x - 3)2

Which answer gives the equivalent factored-form result from removing the binomial GCF that is present among the two polynomial products within 6x(x - 5) - 7(x - 5) ?

(x - 5)(6x - 7)

If an initial factoring arrangement for a "Difference of Squares" is [(x - 3) - (y + 8)][(x - 3) + (y + 8)], then subsequent simplifying of each factor is expected. Simplifying in this case will mean to combine all like terms, limited to those terms in each individual set of brackets. Which response presents the simplified factored form for the given "Difference of Squares"? (Don't forget to fully distribute negative signs when that becomes necessary to eliminate any inner set of parentheses.)

(x - y - 11)(x + y + 5)

Which response illustrates a difference among squared quantities, and is thus a bona fide "Difference of Squares"?

(x-5)2 - (y+3)2

Because it will only be the principal square roots of each term that provide the necessary terms within the binomial factors of a "Difference of Squares", assume then that each variable encountered in this special product represents a nonnegative number. By applying the dependable advice from the previous question along with the requirement that one factor express a difference among terms, while the other express a sum of the same terms, the completely factored form for x6 - y8 is ?

(x3 - y4)(x3 + y4)

Having established the first term of each identical binomial factor for 4x2 + 12x + 9, you now need to secure the second (as well as Last) term for each of these binomial factors. Each of the anticipated identical binomial factors will also end up with the exact same second term; specifically, whatever L is. Commensurately stated, for 4x2 - 12x + 9, where +LL is +9, the last term of each binomial factor will be the principal square root of +9, which gives the term ?

+ 3

Specifically, for the polynomial 9x2 + 12x + 4 to qualify as a "Perfect Square Trinomial": FF needs to be satisfied by 9x2, +2FL has to be fulfilled by the second (or middle) term +12x, and +LL must be embodied by ?

+4

Which is the lowest-degree term within 9x2 - 24x - 4 ?

-4

For the "Perfect Square Trinomial" 36x2 - 84x + 49, assume that F is designated as 6x in order to satisfy for FF being 36x2 . Which value MUST be designated as L, such that BOTH -84x satisfies for +2FL, and +49 satisfies for LL ?

-7

Which selection expresses a completely factored form of the polynomial 8x2 - 18y2 ? (HINT: Follow the six-step "Comprehensive Gameplan for Factoring Polynomials", checking your work as you go.)

2(2x - 3y)(2x + 3y)

Whenever a + or - sign comes between two quantities (sets of parentheses), where each is preceded by a removed GCF as well, it serves to separate one product of polynomials from the next. This exact situation occurs at a stage when factoring by "Grouping". Based on how they are seen here separated by a " - " within 2(x + 5) - x(x + 5), which choice gives both such factored-form polynomial products individually ?

2(x + 5) and x(x + 5)

In light of all qualifying criteria presented in the previous seven questions, which of the following is recognizable as a "Difference of Squares"?

25x4 - 81y8

IMMEDIATELY AFTER a "Perfect Square Trinomial" has been identified with certainty: The factoring of this special product is straightforward, and not difficult once committed to memory. Recall that beyond a GCF, any further-factorable trinomial will A-L-W-A-Y-S result in two binomial factors. Assuming a descending order arrangement among all terms of a "Perfect Square Trinomial", each required identical binomial factor will begin with the exact same term; specifically, whatever F is. Said another way, for 4x2 + 12x + 9, where FF is 4x2, the first term of each binomial factor will include the principal square root of 4 along with a single factor of x, which TOGETHER forms the term ?

2x

...And now the sum of the principal square roots obtained from 4x2 and 9y2 respectively, is given by ________ . (HINT: For each term of the sum, you will need just one of two IDENTICAL, yet comprehensive, factors of 4x2 as well as 9y2 .)

2x + 3y

First, realize that 4x2 - 9y2 is not a trinomial, and second, that it is not a four-term polynomial either. Thirdly, notice that 4x2 - 9y2 has no GCF. At this stage in the factoring process, IT MUST NOT ESCAPE YOUR ATTENTION that this particular binomial is a "Difference of Squares" and thus requires factoring as such. The first binomial factor will be a difference among the principal square roots of each term. The principal square root of the first term, 4x2, is 2x. And, the principal square root of the next term, 9y2, is 3y. So, the response depicting the difference between these two principal square roots is ________ .

2x - 3y

The preliminary factored form for 32x3 - 50xy2, due specifically to the presence of a GCF, is 2x(16x2 - 25y2). However, because 16x2 - 25y2 is a bona fide "Difference of Squares", this only-just-begun factorization of 32x3 - 50xy2 must next be factored equivalently as:

2x(4x - 5y)(4x + 5y)

With regard to the "Grouping" method of factoring: After removing the preferred version(s) of GCF from at least one group (possibly both), it will be important to note the version's resulting + or - sign that is NOT inside a set of parentheses (rather it will be found between them). As an example, such a + sign is found among 2x(x - 3) + x(x - 3). Which selection also features such a + sign?

5(x + 2) + 3x(x + 2)

Therefore, assuming the calculator has been set to allow for decimal places in the answer, a total lack of decimal digits in the displayed principal square root will confirm that the Whole Number that was entered is indeed a perfect square. For instance, in applying the principal square root key to the number 25, a 5 (with no succeeding decimal digits) will appear in the calculator's display window. By contrast, utilizing the same procedure with 26 (instead of 25) will result in 5.0990195, which because of the decimal digits, indicate that 26 is NOT a perfect square. Using a calculator in the manner just described, which of the following numbers is a perfect square?

361

Assuming that (3x3 + 9x2) + (2x + 6) thoroughly checks out as a properly grouped equivalency to a four-term polynomial. The next concern in this overall factoring process is to remove each group's GCF. The response that best depicts what this line of work should look like is ? (HINT: Remember to perform the "Tri-Check" of each GCF.)

3x2(x + 3) + 2(x + 3)

When reading 9x2 - 24x + 40 from left-to-right, which is the FIRST readily obvious reason that this is NOT a "Perfect Square Trinomial"? (HINT: Follow the ordered sequence of all previous questions regarding "Perfect Square Trinomials", which begin with question number 42 .)

40 is not a perfect square itself, and thus cannot satisfy as LL .

So which of these depicts a perfect square TERM? REMEMBER; the numerical coefficient as well as each exponentiated variable, which together constitute a monomial term, must individually be perfect squares!

49x2

Typically (but not always), each of the two factors belonging to a "Difference of Squares" will be a binomial. Which of the expressions below is a binomial?

4x - 7y

In Arithmetic, a product is typically thought of as the result of multiplication. Contrastingly, the factored forms (of polynomials, for instance) are often referred to as products in Algebra. Which selection conveys such an algebraic product?

4x(5x-6)

Let's say that you first check (and ultimately conclude) that both the highest and lowest-degree terms of a trinomial are themselves perfect squares. Because these traits alone do not a "Perfect Square Trinomial" make, the next thing to investigate is this: Does the first-degree term (a.k.a. the middle term of a descendingly-ordered trinomial) have an even-numbered coefficient? If not, then the trinomial will NOT be a perfect square.

4x2 + 5x + 1

If other preliminary factoring steps are necessary (see previous question), do them first. Then decide which of the following does NOT already embody, or whose GCF-less REMAINING factor does NOT qualify as, a "Perfect Square Trinomial"?

4x2 + 6x + 9

Which of the following cannot possibly be a "Perfect Square Trinomial" because either the highest or lowest degree variable terms in x are NOT perfect squares themselves?

4x2 - 12x - 9

Factorable four-term (or more) polynomials will follow their own unique factoring scheme. As a prelude to engaging this plan, you will first need to detect the presence of four (typically, although more are possible) terms, NONE of which are alike. Which of the selections fits this precise description?

4x3 + 8x2 - 3x - 6

As previously stated, not all trinomials will be perfect squares. Thus, IT IS INCUMBENT ON YOU TO DEVELOP THE ABILITY TO RECOGNIZE WHICH ONES ARE. One characteristic that is attributable to a "Perfect Square Trinomial" (assuming descending order for a given variable) is: Both the highest and lowest degree terms within the trinomial must be perfect squares themselves. Based on this trait alone, which choice CANNOT possibly be a "Perfect Square Trinomial"? (FYI: A constant term assumes a zero-degree variable. For example, 3x + 4 is exactly equivalent to 3x1 + 4x0 .)

6x2 - 5x + 9

Noting that 7x4 - 7y10 is neither a trinomial nor a four-term (or more) polynomial, factor this expression by first addressing any GCF concerns. Then perceive whether further factoring is possible in case the disclosed NON-GCF factor is a special product. By following through on this sequence of efficient factoring practices, choose the completely factored form for 7x4 - 7y10 .

7(x2 - y5)(x2 + y5)

A monomial term will only be a perfect square when its numerical (so a number) coefficient AND variable portions are both perfect squares. Which of the following numbers is an acceptable numerical coefficient for a perfect square term? (HINT: Whether it refers to a number or any sort of algebraic expression, a "perfect square" must always be the RESULT obtained from multiplying exactly two identical factors.)

9

In a case where no GCF exists among all terms of a polynomial that needs to be factored, which of the following polynomials could still be factored, specifically because it qualifies as a "Difference of Squares"?

9x2 - 16y4

On the occasion of two different variables being present within a trinomial, then descending order arrangement of terms is based upon just one of these variables. Either of the two variables could be chosen for descending order; the other variable is simply considered as another factor within the terms, and has NO influence on descending order. Despite this available option, for the sake of efficiency: So long as a two-variable trinomial is already given in descending order for one of the variables, there will be one less line of work to provide if you simply leave it that way. Which of the following two-variable trinomials is already presented in descending order for the variable x, and thus does NOT need to be rearranged?

9x2 - 6xy + y2

The variable (letter from an alphabet) portion of any term will only constitute a perfect square when each different variable in the term is raised to a power that is an EVEN number (so a multiple-of-two number). By this guideline, which of the following terms has a variable portion that is a perfect square? (NOTE: Whether the numerical coefficient portion of a term satisfies as a perfect square will be assessed in upcoming questions.)

9x4y2z6

First acknowledge that 7(3x - 2) is one polynomial product (specifically, a product of the monomial factor 7, and the binomial factor 3x - 2 ), and that 2(3x - 2) is another. Which statement(s) is (are) accurate with regard to these polynomial products and their respective factors as they appear together in 7(3x - 2) + 2(3x - 2) ?

All of these

Which of the selections show possible ways of creating two groups of two terms each from 2x3 - 4x2 + 3x - 6, such that there is equivalency with this four-term polynomial?

All of these

Within the +2FL term, there are actually individual monomial factors. The totality of them are: the "automatic" numerical factor +2; a variable factor, such as its numerical coefficient (even if only an implied 1), which together satisfy for F; and one more numerical factor (sometimes accompanied by a different variable, like y), which satisfies for L.

Any of these

Why doesn't 3(64x2 - 9y2) express a completely factored form for 192x2 - 27y2 ?

Because 64x2 - 9y2 needs to be recognized as a "Difference of Squares" and as such must be factored too.

Why DOESN'T 36x2 - 30x + 25 qualify as a "Perfect Square Trinomial"?

Because the middle term -30x, which would need to satisfy as the required +2FL term, doesn't.

In determining whether (x - 2y)(x + 2y)(x2 + 4y2) eventuates as the completely factored form for x4 - 16y4: First see if any of these three binomial factors yet incorporates a (neglected) GCF. If there are no such GCFs, next explore the possibility that one (or more) of these factors is itself a special product. If none of them are, now FOIL only the two binomials which respectively convey the difference and sum of the exact same terms (in this case, these will be x - 2y and x + 2y ) to ultimately obtain x2 - 4y2. Lastly, again to find the product of (x2 - 4y2)(x2 + 4y2). (If you haven't already done so, perform a similar comprehensive check on your answer to the previous question to ensure its equivalency as well as completeness.)

FOIL

Given any trinomial, 4x2 + 16 + 16x for instance, which is always the very first item to address when directed to factor it completely? (HINT: Consult the six-step "Comprehensive Gameplan for Factoring Polynomials" if necessary.)

If it is not yet the exact sequencing of terms, rearrange them to descending order.

For all "Perfect Square Trinomials", assume that FF can only result from identical factors whose numerical coefficients are always the same POSITIVE numbers. Consequent to this assumption, the sign to be included for the ENTIRE term that will satisfy for +2FL would need to be determined from the sign of alone.

L

Does (4x2 - 8y2)(4x2 + 8y2) convey the completely factored form for 16x4 - 64y4 ? (HINT: If it helps you to choose wisely, then perform the factoring on your own using the six-step "Comprehensive Gameplan for Factoring Polynomials".)

No. Although (4x2 - 8y2)(4x2 + 8y2) does give a factored form for 16x4 - 64y4, removal of a GCF was totally bypassed. And directly because of this oversight, another "Difference of Squares", which must ALSO be factored, was never revealed. But, adhering to an efficient sequence of factoring steps would have avoided both of these errors.

So where Real numbers are concerned, if either the highest or lowest-degree terms within a typical trinomial, happen to include a negative number (as either a coefficient or constant term), is there any chance that this trinomial is a perfect square? (NOTE: For present purposes, a typical trinomial is considered to be second-degree.)

No. The highest and lowest-degree terms must themselves be perfect squares, and thus cannot be negative.

Is there a common polynomial factor present in each polynomial product that is separated from one another by the " - " in x(2x + 5) - 4(2x + 5) ?

Yes. The binomial (2x + 5) is a common polynomial factor.

For the trinomial 9x2 + 12x + 4, is it possible for 9x2 to satisfy as FF ?

Yes. The identical factors needed to satisfy for FF are likely 3x and 3x, whose product is 9x2.

Is 9x2 + 12x + 4 a "Perfect Square Trinomial" ?

Yes. The terms of 9x2 + 12x + 4 all individually satisfy for their respective counterparts within FF + 2FL + LL.

Using knowledge gained from the previous few questions regarding squared quantities, which response conveys an accurate initial arrangement of factors, each with the necessary principal square roots, for the "Difference of Squares" given by (x + 7)2 - (y - 6)2 ? (HINTS: Remember that one of these factors, as extensive as it may be, must state an overall difference among each principal square root, whereas the other factor must be the sum of those exact same principal square roots. So make sure you account for this with regard to each of the squared quantities within this "Difference of Squares". Also, because each quantity requires a set of parentheses - or surrogate symbol(s) -, rather than placing one set of parentheses within another, a set of brackets is often used - so maybe here - to "separate" each necessary factor.)

[(x + 7) - (y - 6)][(x + 7) + (y - 6)]

The most common form that perfect square polynomials take is that of a trinomial. This does NOT mean that every trinomial is a perfect square. Nor does it mean that every trinomial, which is able to be factored beyond a GCF, is a perfect square. However, any three-term polynomial whose factors also happen to be two identical factors, WILL BE perfect square trinomials. Because of the type of polynomial factors expected to emanate from a trinomial (disregarding any GCF), such identical factors could only be ?

binomials

Although other justifications for doing so exist, arranging terms in descending order makes it easier to find the highest and lowest- terms in polynomials consisting of more than two terms. Because these highest and lowest-terms have added importance when factoring trinomials, this specific type of polynomial is to be written in descending order at the start of the factoring process.

degree

What you should have learned from the prior thirteen questions is that certain trinomials cannot possibly be . Despite initial and secondary characteristics that help to recognize the possibility of this special product, a final confirmation yet awaits. Toward this end, you will come to realize that an in-depth knowledge of the workings of FOIL will prove most useful.

perfect squares

Just as certain numbers such as 1, 4, 9, 16, etc. are perfect squares, certain polynomials have the possibility of being perfect squares as well. The reason that a number like 25 is a perfect square is because IT IS THE PRODUCT OF TWO I-D-E-N-T-I-C-A-L FACTORS; 5 x 5 in this case. With regard to perfect square numbers, these identical factors are not unexpectedly, numbers as well. In a similar way, any polynomial that happens to also be a perfect square must then have two identical factors itself. However, rather than being numbers, the identical factors of a perfect square polynomial must always be other ?

polynomials

A trinomial will ONLY be a perfect square if its terms, arranged in descending order (for a single variable), fit the pattern FF + 2FL + LL . (Note that within this three-term arrangement, FF (a.k.a. F x F) and LL (a.k.a. L x L), which respectively begin and end this pattern, are both perfect ; FF depicts a product consisting of two identical factors of F, just as LL consists of two identical factors of L.)

squares

So to help recognize a polynomial that satisfies as a "Difference of Squares", how many items (numbers, monomial terms, algebraic quantities, or any combination thereof) will you likely need to account for in a polynomial?

two

Although further work will be necessary to achieve completeness, principal square roots of each term, quantity, or other expression which embodies a perfect square, will always be required when factoring ANY "Difference of Squares". Also, when factoring a polynomial, unless stated otherwise, it is ordinarily assumed that each contributory variable symbolizes a nonnegative number. Under these usual conditions, which response gives the principal square root of (x + 5)2 ? In other words; because the particular whole number exponent is 2, the expression (x + 5)2 suggests two identical factors of the quantity (x + 5). So select just one of these identical factors to obtain the requested root.

x + 5

If a numerical exponent is a multiple of two (so, an even number), then at least the variable portion of a term such as x6 will be a perfect square. Assuming that the variable represents a nonnegative number, the principal square root of all such terms is easily obtained. Simply divide the original exponent by two (specifically because of it being a multiple of two) and use the resulting number as the new exponent. So which of the selections gives the correct response for the principal square root of x6 ?

x3