Math MC Quiz
A polynomial that is prime may synonymously be characterized as .
unfactorable
Based SOLELY upon the greatest quantity of factors, which of these most likely presents the completely factored form for 2x2 + 12x + 16 ?
2(x + 2)(x + 4)
Due to the expectation of factoring trinomials completely: The very best way to proceed (assuming the trinomial is already in descending order) is to always remove (factor out) any GCF before something else! Which of the responses accomplishes this (not to be overlooked) step in the process of factoring 2x2 - 4x - 30 ?
2(x2 - 2x - 15)
Which response lists as many factors as possible for the monomial (single-term polynomial) 6xy ?
2, 3, x, y
Thus, the completely factored form of 6xy is given by:
2·3·x·y
Displaying the result of factoring out (removing) a GCF from all terms in a polynomial typically begins with the GCF itself, followed immediately by: a collection of each remaining factor from each term of the initial polynomial. Such a collection consists entirely of the remaining factors of those terms, now expressed as "new" terms within a single set of parentheses (or any nonnumeric symbols - like a horizontal fraction bar - which would imply parentheses). Thus, the conventionally correct and complete way of showing the removal of the GCF from 3x + 6 is .
3(x + 2)
Which response lists the most factors possible for the monomial 15x2y ?
3, 5, x, x, y
Among all possible factors for 6xy and 15x2y respectively, which response lists the most factors that are exactly the same for both 6xy and 15x2y ?
3, x, y
The binomial, 6xy + 15x2y, can be accurately perceived as the sum of the 6xy and 15x2y terms. Which response lists the most factors possible that are exactly the same for both of these algebraic terms?
3, x, y
The completely factored form of 3x - 12 is 3(x - 4), where the removed GCF is 3. Which response is obtained as a result of distributing 3 over (x - 4) ?
3x - 12
So which response is the GCF of 6xy + 15x2y ?
3xy
Which response gives the PRODUCT obtained from the most factors possible that are also exactly the same for both terms found in the binomial 6xy + 15x2y ?
3xy
Which response gives the collective PRODUCT of individual component factors that are all common to every term within the polynomial 6xy + 15x2y ?
3xy
Finally, if you were asked to factor out the GCF from 24xy2 + 3x2y - 18xy so as to satisfy the guideline regarding positive or negative GCFs, which response fully complies?
3xy(8y + x - 6)
So if you were asked to factor out the GCF from 3x2y - 18xy + 24xy2, which response is in accordance with the guideline stated in question 64?
3xy(x - 6 + 8y)
The first of the three parts of the 'Tri-Check' is to make sure that the same number of terms (from among the collection of remaining monomial factors) actually appear inside the parentheses as that which the given (initial) polynomial had. Then, presuming that the GCF has been removed, as the second part of this check: Examine each term within the remaining collection of terms. If any common factor yet exists among all these terms, then the Greatest Common Factor has not really been established. Which of the choices has NOT had a true (all-inclusive) GCF removed because a common factor is still present among those terms which form the collection of remaining terms within the parentheses? (HINT: See if any prime number - especially among 2, 3, and 5 - has been an "overlooked" common factor.)
4(2x2 - 6x + 8)
Where polynomials are concerned, a coefficient of 1 is typically not required to be physically present as a factor; it is simply assumed. Thus, if asked to give the written factored form of a polynomial that possesses no other possible factors than itself and 1, such as 4x + 9, you would likely write:
4x + 9; unless specifically directed to also use 1, or where the inclusion of 1 would be contextually indispensable for the factored form of this expression.
Given that 15x2 - 30x - 45 is to be factored completely, what common factor continues to exist among all terms within the parentheses of 3(5x2 - 10x - 15) ?
5
Given that 5x(2x2 + 3x - 4) is the completely factored form of a polynomial product, list the individual polynomial factors for this product. (NOTE: Because it would be counterproductive to that which is desired here, do NOT distribute.)
5x, and (2x2 + 3x - 4)
Which monomial gives the PRODUCT of the most factors possible that are also exactly the same for ALL terms found in the trinomial 5x3y - 10x2y2 + 15xy2 ?
5xy
Which of the following factored forms best facilitates further factoring of 8x2 - 32y2, assuming that none of the responses yet portrays a complete factorization?
8(x2 - 4y2)
Is it possible then, for the factorization of a monomial to consist of contributory factors that are themselves monomials ?
Absolutely. In fact, all factors of any monomial will always consist solely of monomials.
Consider that the result of multiplying 2·5·x·y·y gives 10x2y2. What specific types of polynomials are 10x2y2 , 2, 5, x, and y respectively?
All of them fit the definition of a monomial.
Which response describes the essence of a completely factored polynomial ?
All of these.
Within the specific relationship between factors and a product, how does one use factors (where all such factors are Rational Numbers and/or polynomials) to obtain the equivalent product?
Multiply all factors together
Based on what you have hopefully deduced from the previous four questions, which selection most likely states a universal truth regarding the relationship between a polynomial and its factors?
No single polynomial factor exceeds the degree of its polynomial product.
In example after example, a universal truth regarding each number considered specifically as a factor of some positive numerical product, is:
No such number could ever be greater in value than that positive numerical product.
Is (x - 3) a factor within the following expression: (x - 3) + (x + 5)?
No, for ANY reason given in the other selections here.
Will it ever be SUFFICIENT to verify the completeness/correctness of removing a GCF from a polynomial by simply distributing the "supposed" GCF over the collection of remaining terms?
No. Distributing by itself (so without the benefit of the previous two parts of 'Tri-Check') does not provide a foolproof way to ensure the all-inclusive nature, with regard to all factors in common with all terms in a collection, of a GCF.
One set of factors for the number 12 is given by 4·3. Assuming that they are unique in comparison to the number given (so 12 here), must there be at least two OTHER numbers (like 4 and 3) to satisfy as a set of factors for any given number?
No. For instance, the only set of factors for the number 17 are itself (thus, not unique) and 1. In other words, 17·1 gives the exclusive set of factors for 17.
Are either x - 2 or x2 + 2x + 4, which are factors of x3 - 8, polynomials of greater degree than x3 - 8 ?
No. The binomial x - 2 is first-degree, and the trinomial x2 + 2x + 4 is second-degree, whereas x3 - 8 is a third-degree binomial.
Are either 2x or (x3 + 4x2 - 2x + 3), which are factors of 2x4 + 8x3 - 4x2 + 6x, polynomials of greater degree than 2x4 + 8x3 - 4x2 + 6x ?
No. The monomial 2x is first-degree, and x3 + 4x2 - 2x + 3 is a third-degree polynomial, whereas 2x4 + 8x3 - 4x2 + 6x is a fourth-degree polynomial.
The polynomial, 9x2 - 81y2, can equivalently be factored to give (3x - 9y)(3x + 9y). However, does (3x - 9y)(3x + 9y) represent the complete factorization of 9x2 - 81y2 ?
No. There are GCFs among the terms of each binomial, which would not exist had GCF concerns been attended to prior to the factored form expressed above.
...So, can 3 be the GREATEST Common Factor of 15x2 - 30x - 45 ?
No; besides 3, there is at least one other factor that is common to the given three terms.
Are either (x - 2) or (x + 4), which are factors of x2 + 2x - 8, polynomials of greater degree than x2 + 2x - 8 ?
No; each of these binomial factors is first-degree, whereas x2 + 2x - 8 is a second-degree trinomial.
Two arrangements which convey the completely factored form of the polynomial 2x - 5 are 1(2x - 5) and (2x - 5)1. Yet for reasons other than being in compliance with the specific request for a comprehensive set of factors, is there any value-wise advantage whatsoever in using either 1(2x - 5) or (2x - 5)1 instead of simply 2x - 5 ?
None that is apparent; each of these iterations conveys just one consequential factor, which is 2x - 5 .
Consider the number 24 as a product. The exhaustive list of pairs of natural number factors for 24 is: 1&24, 2&12, 3&8, 4&6, 6&4, 8&3, 12&2, and 24&1. Which is true regarding any of these numbers that are factors of the product 24?
Not one of them exceeds the amount of the product, 24.
Whether given in pairs or not, which will be true regarding all Integer Numbers that comprise the factors for the product 36?
Not one of them is a number whose amount exceeds the product, 36.
Every single term of a polynomial is itself a product of factors. For instance, within the binomial 5x + 5, the term 5x is the product of 5·x, and the term 5 is the (assumed) product of 5· .
1
Recall that for a variable term such as x, there is an understood "attached" numerical coefficient of 1. In other words, x really means 1 · x or simply 1x. In the same way, when specifically written as 3x + 8 the same factor (or coefficient) , must presumably exist in order to consider 3x + 8 as a product (of 1 and itself).
1
The GCF of the binomial 5x + 5 is the monomial 5. If the GCF is removed as a factor from the term 5x, then the only factor remaining from this term is x. Likewise, if the GCF is removed as a factor from the term 5, then the only remaining factor of 5 is .
1
REMEMBER: It will (very) often be helpful to first consider descending order prior to factoring any trinomial. So, with regard to the descending order for -6 - 2x + 10x2, which will be the GCF?
2
So which of the following depicts (x + 1) as a factor?
(x + 2)(x + 1)
Which of the following depicts a product?
(x + 2)(x + 1)
Both (x2 - y2)(x4 + x2y2 + y4) and (x - y)(x + y)(x4 + x2y2 + y4) are factored forms of the polynomial x6 - y6. Which of these two sets of factors most likely expresses 2x2 + 12x + 16 in its completely factored form?
(x - y)(x + y)(x4 + x2y2 + y4), because it has more factors; two binomial factors and one trinomial factor.
Which collection of terms remains after removing a GCF of 3 from 3x2 + 9x - 15 ?
(x2 + 3x - 5)
Consider the collection of terms comprised of each REMAINING factor from each term within a polynomial after the GCF has been removed. As the "new" terms (in this collection), each must begin with either a or sign, which serve as the SOLE way of separating one term from the next in any algebraic expression.
+, -
In reference to the previous question, if the GCF is changed to -3, and it is removed from the same -9x, then the resulting "new" term in the collection must now be .
+3x
From another perspective then, the act of factoring out (removing) the GCF from every term within a polynomial is essentially the same as dividing out the GCF from every term of the polynomial. For instance, removing the GCF, -2, from -2x2 + 6x - 8 achieves the same result as simplifying
-2, -2, -2
But, if you were now asked to factor out the GCF from -18x2y + 3x2y + 24xy2, which response will satisfy the guideline for determining whether the version of the GCF is positive or negative?
-3xy(6 - x - 8y)
In removing the GCF from 4x - 8y, the completely factored form will be 4(x - 2y). Yet, say the same polynomial had its terms rearranged so that it was presented as -8y + 4x instead. Because the lead term has changed and because this term now displays a NEGATIVE numerical coefficient, the completely factored form will now more suitably be . (Note: Be sure to distribute to validate equivalency with -8y + 4x .)
-4(2y - x)
Although not necessarily performed 'completely', various factored forms for 2x2 + 12x + 16 include the two binomial factors (2x + 8)(x + 4) and (x + 4)(2x + 4) respectively, as well as 2(x + 2)(x + 4), which consists of one monomial factor and two binomial factors. Which of these exhibits the greatest quantity of factors?
2(x + 2)(x + 4)
Which response displays descending order for -6 - 2x + 10x2 ?
10x2 - 2x - 6
Given that 5x(2x2 + 3x - 4) is the completely factored form of a polynomial product, which is the polynomial that this product must have been? (Hint: Use the final part of 'Tri-Check'; so carefully distribute the GCF, 5x .)
10x3 + 15x2 - 20x
The directive to factor a polynomial will nearly always mean to factor it completely. The GCF plays a major, often primary role in accomplishing this. Which of the responses depicts 12x - 24y in its completely factored form?
12(x - 2y)
Upon removal of the GCF, which response depicts the resulting completely factored form for 15x2 - 30x - 45 ? (Hint: Make use of the 'Tri-Check' to verify completeness as well as equivalency.)
15(x2 - 2x - 3)
Because of it being all-inclusive of every factor common present in each term among 15x2 - 30x - 45, the GREATEST Common Factor for this trinomial is ?
15; which includes the individual numerical prime factors of 3 and 5
Which response(s) give(s) a satisfactory common factor for all the terms among 6xy + 15x2y ?
Any of these
Which colossal blunder(s) may potentially occur if a GCF is not removed at (or near) the outset of factoring any polynomial?
Both B and C
An equivalently factored form of 2x2 - 4x - 30 is (2x + 6)(x - 5). However, it does not give the completely factored form for 2x2 - 4x - 30. Which of these responses provide an explanation as to why this is so?
Either the A or C repsonses are satisfactory.
Because of the primary role a(n) often has in factoring polynomials completely, it is essential that any mistake made in the process of removing a(n) be corrected IMMEDIATELY. To ensure that the resulting factored form is both complete, and equivalent to that of the initial polynomial, a 'Tri-Check' (three-step check of) MUST subsequently be implemented whenever removing a(n) ?
GCF
The third (and final) part of the 'Tri-Check' for completeness/equivalency in removing a GCF involves distributing. Recall that distributing occurs when multiplying a single value over (to) all terms within a collection, such as found within parentheses. For purposes of the aforementioned 'Tri-Check', what single value must be distributed?
GCF
Upon having removed (factored out) the GCF from a polynomial, additional factoring is often necessary in order to satisfy factoring completely. Any such consequent factoring WILL A-L-W-A-Y-S BE EASIER TO ACHIEVE SIMPLY BECAUSE THE HAD ALREADY BEEN REMOVED.
GCF
Within the context of Algebra, the abbreviation GCF stands for.
Greatest Common Factor
Another equivalently factored form of 2x2 - 4x - 30 is 2(x2 - 2x - 15). What was done to achieve this form?
The GCF was removed from each term of the given trinomial.
How does distributing help to check whether the removal of a potential GCF from a polynomial has been done accurately, even if not completely?
The product that results from distributing a suspected GCF over the collection of remaining terms must result in the exact polynomial as it was prior to having removed the GCF, thus establishing necessary equivalency with the given (initial) polynomial. If no such equivalency is established, then the process of removing the GCF could not have been done correctly.
Regardless of the previous question, are there any practical differences in values if expressing 4 - 9x as 1(4 - 9x) instead?
They both convey the exact same value. In either case, the only factor of consequence is 4 - 9x.
REMEMBER: Perform the " - Check" whenever removal of a monomial GCF from a polynomial is either required or helpful. Each part of this check is applied in the following order: 1) By matching the number of terms that are placed within parentheses to those of the given polynomial, then 2) By checking that it really is the GREATEST of Common Factors which has been removed; by seeing that no other common factor among terms now within the parentheses yet exists, and 3) By distributing the GCF to obtain the original polynomial. Only when you adhere to this practice, will BOTH aspects of completeness and equivalency, regarding the GCF, be guaranteed.
Tri
So, as it relates to any Natural Number: The phrase "factor completely" means to create a factored form such that as many factors as possible are given, yet whose comprehensive product results in that very same Natural Number.
True. And by the way, there is no way to establish a product with as many factors as possible other than to use prime factors only.
Within the scope of Natural (counting) Numbers, 1, meaning one entire thing, is perceived as UNITY and thus, unlike any other Natural Number, is not an amount that you could subdivide in at least one way to result in at least one OTHER Natural Number. For instance, 1 can only be subdivided into 1, whereas 6 can be subdivided into 2 or 3. It is for this very reason that 1 is not included among the prime numbers.
True. And by the way, unless there is a specific directive or useful reason to include 1 in the factored form of any polynomial, don't!
Because descending order arrangement is highly preferable when specifically factoring any trinomial, whether the GCF should be positive or negative must be considered within the context of this particular succession of terms. For instance, a case could be made that -4 is the proper GCF for the polynomial -16 + 4x + 36x2. Being mindful of the descending order of this trinomial first, the same polynomial now presented as 36x2 + 4x - 16, would need a GCF of +4 instead.
True. Because it is typically done prior to factoring trinomials, rearranging to descending order may provide a lead term for this type polynomial which often alters the sign that the GCF might have otherwise been.
In the same way that the only set (pair) of factors for some numbers consist entirely of that same number and 1, the full complement (often a pair) of factors for some polynomials may also consist of only that polynomial and 1.
True. Considering that polynomials are algebraic representations of numbers, it stands to reason that some of them could be prime too.
The identity element of multiplication is 1: If any number, value, or expression is either multiplied by 1 (or multiplies 1), the resulting product is simply that number, value, or expression respectively.
True. For instance, 1·9 and 9·1 both result in 9, and 1(3x + 7) and (3x + 7)1 both result in 3x + 7.
A conclusion that can be made from the information presented in the previous ten questions is: If the polynomial is not suitable for grouping, and where descending order has already been fulfilled for any trinomial if given, the very next item of business in factoring this polynomial is to see whether there is a GCF present that needs removing.
True. Skipping this step could have devastatingly detrimental effects.
For all intents and purposes, as it relates to any polynomial: The phrase 'factor completely' means to present a factored form such that as many polynomial factors as possible are utilized, yet whose comprehensive product results in the polynomial that was given.
True. There is no way to establish a product with as many different factors as possible other than to use factors which are exclusively prime.
Whether an individual "new" term within the collection of terms, which remains after the removal of a GCF, begins with one sign or the other depends upon two things: the sign used for the GCF itself, and that sign of the given polynomial's "old" term PRIOR to removing the GCF. For instance, if the GCF is 3 and it is removed from -9x, then the resulting "new" term in the collection will be -3x .
True. When a positive factor like 3 is removed from a negative product such as -9x, then the resulting "new" term will maintain a negative (numerical) coefficient to satisfy a distributing (multiplication) check.
In light of the existence of an identity element of multiplication, if asked to give the complete list of all possible factors for any number, value, or expression must 1 be included in this list?
Yes, but this is done mainly for the sake of "completeness" with regard to the specific question. Yet, multiplying by a factor of 1 will always return an identical number, value, or expression.
Is 3 a common factor among all terms of the polynomial 15x2 - 30x - 45 ?
Yes; 3 is a factor within 15 and 30, which are the numerical coefficients of the leftmost two terms of the polynomial. And, it is also a factor of the constant term 45.
The same monomial that satisfies as a factor among every single term within a multi-term polynomial, routinely describes a .
common factor
Both 2(x2 - 2x - 15) and (2x + 6)(x - 5) are simply two possible factored forms of 2x2 - 4x - 30, much as 2·6 and 3·4 are possible factored forms for 12. However, whether discussing numbers like 12 or polynomials such as 2x2 - 4x - 30, there can ultimately be only one (ignoring oppositely signed versions of the GCF) factorization of either.
complete
AN ISSUE TO BE AWARE OF with regard to any polynomial is: If the instruction simply states "Factor the polynomial", you will still be expected to factor it .
completely; all prime factors
Because division reverses multiplication: Displaying the removal of a monomial GCF from a polynomial, along with the collection of remaining terms, exactly conveys the reversal of the process known as ?
distributing
One way to compare polynomials is to consider the 'greatness' of each with regard to their respective degrees. If compared this way, the automatic inclination would be to perceive a -degree polynomial as greater than a fourth-degree polynomial.
fifth
A factored form for the polynomial 2x4 + 8x3 - 2x4 + 6x is: 2x(x3 + 4x2 - 2x + 3). These factors happen to be -degree and third-degree polynomials respectively.
first
A factored form of the trinomial x2 + 2x - 8 is (x - 2)(x + 4). Each of these two factors are themselves binomials that also happen to be -degree polynomials.
first
Consider a single monomial that satisfies as a factor of every term within a multi-term polynomial. Consider further that this particular monomial is also the collective product resulting from ALL possible individual component monomial factors that are the SAME within every term of this multi-term polynomial. (For instance, if each term among a polynomial all had 2, 3, 5, x, and y as factors, then the single monomial 30xy would be this collective product.) The description provided here is for more than merely any common factor. Rather, it specifically describes a common factor.
greatest
Within the context of polynomials, each factor that is presented in the completely factored form for 6xy is specifically considered to be a factor.
monomial
A factor is any algebraic expression or number that interacts with any other algebraic expression(s) or number(s), specifically via the operation of in order to form a single product.
multiplication
Where GCFs are concerned, exactly two versions (options) will always be possible. We will refer to these as either a positive GCF or negative GCF. Comparatively then, the ONLY deviation between them is that each is signed relative to the other.
oppositely
There will be future considerations, based upon new material, in deciding which version of the GCF will be more efficient to use. For now, a consistently useful guide to follow is: Examine the numerical coefficient of the lead term of the polynomial to be factored. If that coefficient is a negative number, then the GCF should begin with a negative number as well. (Note: Not necessarily the exact same negative number, depending on other relevant aspects of the Greatest Common Factor.) However, when the numerical coefficient of a polynomial's lead term is positive, then begin the GCF with a number instead.
positive
Considering that the only available set of factors for each of the following numbers: 2, 3, 5, 7, 11, 13, and 17 is the number itself along with 1, these and any number having the exact same characteristics are defined as .
prime
If directed to completely factor a (Natural) number, expect to show a series of multiplications, involving factors only, such that the product of all these multiplications result in the initial (Natural) number. For instance, although it is possible for 60 to be expressed in a factored form given by 6·10, the completely factored form of 60 MUST instead be expressed as 2·2·3·5, or equivalently as 22·3·5.
prime
If the only possible set of factors for the binomial 2x + 5 are itself and 1, then as a factor, whether a polynomial or not, 2x + 5 is considered to be.
prime
It is useful to keep in mind that similarities of many principles exist among the processes of factoring Natural Numbers and factoring polynomials. Just as with any Natural Number, for any polynomial then: 'Factoring completely' will mean to establish a series of multiplications, each involving factors only, such that the product of all these multiplications result in the initial polynomial.
prime
With regard to 'factoring completely', the only difference between doing so to Natural Numbers or to polynomials is: Factors of a Natural Number will also be Natural Numbers, whereas factors of a polynomial will need to be polynomials (so NOT numbers). However, in both cases, ALL factors must be.
prime
Essentially, composing a completely factored form of a Natural Number means the exact same thing as expressing that number's ?
prime factorization
Consider that the only factors of a certain polynomial are 1 and that same polynomial. In a majority of situations where it might be needed, the written factored form of this polynomial will not include the factor of 1. Because this factorization of such a polynomial will thus display no difference at all from itself, it is concluded that this polynomial is essentially .
unfactorable
The GCF of the binomial, 3x + 6, is the monomial 3. Upon removing the GCF from both terms of this binomial, what factors will remain for each term RESPECTIVELY?
x and 2
Which of the selections below depicts x2 as a factor?
x2(x3 + 4y)
