Algebra II Mid Term Exam Study Guide
What are the solutions to 36x^2 = 18x?
0.5 0
Which equation is part of the correct solution?
c.) (x + 4)^2 = 30
Is the leading coefficient positive or negative?
positive
Factor the expression
2(x-5)(x-6) -2(x-5)(x+6)
what is the vertex of the parabola formed by this equation? (___, ___)
( -3, -25)
Which quadratic expressions name functions that, when graphed, will cross the x-axis at -1 and 5? Select all that apply.
(2x^2 -8x -10) (x^2 -4x -5)
Use synthetic division to determine if (x - 1) is a factor
(x - 1) is not a factor
Consider the function. Use + ∞, or - ∞.
+ ∞ + ∞
Match each expression to the vocabulary term that describes it. Although some expressions can fit more than one vocabulary word, every vocabulary word will only be used once and only once. Monomial, binomial, polynomial, trinomial, constant, leading coefficient, constant
- Binomial - Monomial - Polynomial - Trinomial - Leading coefficient - Coefficient - Constant
Which of the following is true about the parabola? - The parabola has x-intercept as ( 0, 0 ) and ( -4, 0 ). - The parabola can be represented by the equation Y = -(x-2)^2 + 4 - The parabola has an axis of symmetry of x = 2. - The y-intercept of the parabola is the origin. - The vertex of the parabola is at ( -2, 4 ).
- The parabola can be represented by the equation Y = -(x-2)^2 + 4 - The parabola has an axis of symmetry of x = 2. - The y-intercept of the parabola is the origin.
Complete the statements below by choosing "always", "sometimes", or "never" to create true statements. - The terms of a polynomial are ______ monomials. - The sum of two trinomials is _____ a trinomial. - Like terms ______ have the same coefficient and the same variable factor. - Subtractions of a term can ______ be rewritten as addition of the opposite term. - A binomial is ______ a polynomial of degree 3.
- The terms of a polynomial are always monomials. - The sum of two trinomials is sometimes a trinomial. - Like terms sometimes have the same coefficient and the same variable factor. - Subtractions of a term can always be rewritten as addition of the opposite term. - A binomial is sometimes a polynomial of degree 3.
Which of the following statements are true? Select all that apply.
- The x-intercept of the parent function has been shifted left 2 units. - domain of the translated function is all real numbers. - On the interval -5 ≤ x ≤ 0 from left to right, the function increases only.
Madeline divided a polynomial by a polynomial and got a negative remainder. Is this possible or does Madeline need to rework the problem? - This is not possible and Madeline should rework the problem. - This is possible and no mistakes were made.
- This is possible and no mistakes were made.
- What is the relationship between the degree of each of the previous equations with the number of roots? - How do the graphs of the related functions help classify the roots of the equations? - How does factoring the related polynomial expressions help classify the roots of the equations?
- with the 1st expression, it has a degree of 4, and 4 roots. So does the 2nd expression, the 2nd expression has a degree of 4 and has 4 roots. The last expression has a degree of 4 but only has 2 roots. - The graphs of the related functions help to classify the roots of the equations by crossing the x-axis which determines what the roots of the functions are. - By factoring the related polynomial expressions, you can determine the roots of the equations by setting the equations equal to zero.
In the graph below, complete the following end behavior. Use - ∞ or + ∞
- ∞ - ∞
Using synthetic division, select the rational roots from the list of possible rational roots below. -1 -2 -4 8 4 1 2 -8
-1 4 2
Identify each of the following parts of this expression as a constant term, variable term, or coefficient.
-4.9: coefficient -4.9t^2: variable term 22t: variable term 22: coefficient 1.8: constant term
Identify the category that describes the end behaviors of each of the polynomial functions.
1. b 2. d 3. a 4. c
match the letter to the equivalent expression number
1. c 2. a 3. b 4. f 5. d 6. e
Identify the types of roots for a quadratic when the discriminant
1. d 2. c 3. a 4. b
Determine the factors. Select all that apply. 2 3 -3 4 -2 0
2 -3 0
If x − 5 is a factor of this polynomial, what is the value of a?
22
Find the quotient (and remainder if applicable) of
2x + 1
Divide
2x + 1 r. 2 or 2x = 1 2/x - 3
Factor and make a conjecture.
2x^2 - 7x -4 = (2x + 1) (x - 4) 2x^4 - 7x^2 - 4 = (2x^2 + 1) (x^2 - 4) I can factor out these 2 polynomials by splitting the middle term. They both end up in parentheses with similar terms. The only difference is the 2nd expression has variables to the 2nd power.
Which of these expressions is equivalent? Select all that apply.
3t^2-t -4 (3t -4)(t + 4)
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: )
link
:)
Fill in the values for each missing letter. A = B = C = D =
A = -4 B = -28 C = -5 D = -20
Analyze the behavior for each function representation. Choose all functions that represent each statement. If no function matches, then select none. Options: k(x), j(x), h(x), or none
A. k(x) B. j(x) C. j(x) D. none E. k(x) & j(x) F. h(x) & j(x) G. none
Divide: identify the quotient and the remainder
Quotient: x^2 + 2x + 4 Remainder: 0
Which functions have a negative leading coefficient? Select all that apply.
B D
Fill in the blanks. The polynomial expression has ________ terms, so it is a _______. Blank 1: - 5 - 4 - 2 - 1 - 3 Blank 2: - Trinomial - Binomial - monomial
Blank 1: 3 Blank 2: trinomial
Use synthetic divisoin to determine if x - 3 is a factor. Use the choisces to complete the sentences. Using synthetic division, the top row of numbers is (Blank 1) and the potential root is (Blank 2). After dividing, the result is (Blank 3). This means that x - 3 (Blank 4) a factor of P ( x ) since the remainder is (Blank 5).
Blank 1: B Blank 2: B Blank 3: D Blank 4: A Blank 5: D
How could you use the factored form of the polynomial to determine the roots of the equation
By using the factored form of the polynomial and setting the expressions to zero I can determine that the roots of the polynomial is -3 and 3.
What is the factored form and the standard form of f(x) ?
Factored Form: f(x) = (x - 4)(x + 4)(x + 1) Standard Form: f(x) = x^3 + x^2 - 16x - 16
How many real roots and how many complex roots would you expect the equation x^4 − 16 = 0 to have? Explain your reasoning.
I know there would be a total of 4 roots since the degree is 4. There would be 2 real roots and 2 complex roots after factoring out the equation. By completely factoring out the equation you will find that the two real roots are x= 2, x=-2 and the complex roots are x=2i, x=-2i.
Place the steps in the correct order. x² + 6x + 9 = 40 + 9 (x + 3)² = 49 x = 4; x = -10 x² + 6x = 40 x+3 = ± 7
Step 1: x² + 6x = 40 Step 2: x² + 6x + 9 = 40 + 9 Step 3: (x + 3)² = 49 Step 4: x+3 = ± 7 Step 5: x = 4; x = -10
The equation of the axis of symmetry is x = ? The vertex is at (___, ___)
The equation of the axis of symmetry is x = -2 The vertex is at ( -2, 5 )
Use "odd" or "negative" The function's degree is ______ and its leading coefficient is _________.
The function's degree is odd and its leading coefficient is negative.
Recall that multiplicity is how many times the root or factor appears. What is the multiplicity of each root? How do you know?
The multiplicity of each root is 2 because the root appears two times.
The value of p(4) is (Blank 1). Based on this result, (Blank 2) (Blank 3) a factor.
The value of p(4) is 165. Based on this result, x - 4 is not a factor.
Use the remainder theorem to calculate the value of p(4). Therefore, by the remainder theorem, p(4) = ____, and the remainder after dividing (equation) by x - 4/ x +4 is 26, -26, 24, -24.
Therefore, by the remainder theorem, p(4) = 26, and the remainder after dividing (equation) by x + 4 is 26.
Relate the factoring of a polynomial to polynomial division by filling in the blanks in the sentence below using dividend, remainder, or divisor or quotient. When there is no remainder in polynomial division, the ________ and the _______ are factors of the _______.
When there is no remainder in polynomial division, the divisor and the quotient are factors of the dividend.
Y-intercept: the maximum point of the parabola: the minimum point of the parabola: Zeros: Translation from y = ax^2:
Y-intercept: standard form the maximum point of the parabola: vertex form the minimum point of the parabola: vertex form Zeros: standard form Translation from y = ax^2: vertex form
Completely factor
a.
Determine the roots of the polynomial.
a.
Factor the expression completely.
a.
Rewrite the expression. Combine all like terms. Your final expression should not contain any parentheses.
a.
Select the choice with the correct degree, leading coefficient, and end behavior
a.
State the maximum number of turns the graph of f(x) could make.
a.
What are the factors of the expression?
a.
What is the G.C.F of the terms of the expression
a.
Which is true about the polynomial function?
a.
Which statement is not true about this function?
a.
Select the expressions that contain representations that are equivalent to the difference of perfect squares. Select all that apply.
a. d. e. g.
Select the factors that are part of the polynomial
a. b.
Solve for the real zeros of the function
a. c.
Which functions will have the type of roots described? 2 real 1 complex, 3 complex, 3 real, 1 real 2 complex
a. 3 real b. 3 real c. 1 real 2 complex
Match each division expression with the correct quotient.
a. 5 b. 1 c. 6 d. 2 e. 4 f. 3
Which of these polynomial expressions have a degree of 2? Select all that apply.
a. and c.
Solve 3x - 6x^2 = -9
a.) x = -1; x = 3/2
Divide
b .
Divide
b.
Expand
b.
Factor the cubic polynomial completely.
b.
Factor the expression
b.
Find the remaining zeros given that -3 is a zero of f(x).
b.
Rewrite p ( x ) as a product of its linear factors.
b.
State the maximum and minimum number of zeros for f(x).
b.
What could be the equation for this graph?
b.
Which function best matches the function shown:
b.
Which statement is true based on the graph below?
b.
divide
b.
expand the polynomial expression:
b.
Select all the graphs which show even degree functions.
b. c.
Determine the factors. Select all that apply.
b. c. e.
What are the roots of the function? Select all that apply.
b. d.
The equation of a polynomial with a root of multiplicity 2 at x = -3 and a root of multiplicity 1 at x = 0 is: select all that apply
b. f.
Which of the following expressions show the difference of two perfect squares? Select all that apply.
b. c.
What is the y-intercept of the following cubic function?
b. (0, 2)
Which of the following polynomials are equivalent to each other.
b. and c.
Which graph shows the line 2x - 5y = 15?
b.)
Factor
b.) (3y - 2)(3y - 2)
Which of the following expressions could have been one of the factors Lin used to find this solution?
b.) 4x-3
What are the solutions to Amanda's equation?
b.) x = -5; x =4
What are the solutions to (x + 3)^2 + 4 = 29
b.) x = 2; x = -8
A cube has edges that measure 3x + 3y. Which of the following expressions is equivalent to the cube's volume?
c.
Completely factor
c.
Determine the x-intercepts of K ( x ) and then rewrite K ( x ) as a product of all its factors.
c.
Divide
c.
Factor the cubic polynomial completely.
c.
Factor the expression by applying the identity for the sum of two cubes.
c.
Factor the expression completely.
c.
Factor the following expression:
c.
State the minimum degree of the function shown:
c.
Use long division to determine the quotient
c.
What are the zeros of the polynomial
c.
Which statement is true based on the graph above?
c.
Write the polynomial expression in expanded form.
c.
factor
c.
he perimeter of a flower garden is 54 feet. Its length is 6 feet more than twice its width. Let the width of the garden be represented by x. Which equations can be used to describe this situation?
c.
Joan wants to factor the quadratic expression by using the graph of the related quadratic function below and identified the x-intercepts of the graph to help with her factoring. Use the graph to write the expression in factored form.
c. (x - 2) (x + 4)
Determine the roots of the polynomial.
d.
Factor the expression by applying the identify for the sum of two cubes.
d.
Find the zeroes of f(x).
d.
State the maximum number of turns the graph of f(x) could make.
d.
What are the zeros of the function? Use factoring to help you find the answer.
d.
What is the y-intercept of f(x).
d.
Which choice best describes the zeros?
d.
Which is true about the polynomial function?
d.
Write the polynomial expression in expanded form.
d.
divide
d.
Select the factors that are part of the polynomial
d. e. f.
Be sure to "rebuild" the polynomial in factored form, from the numbers in the synthetic division process. Select all that apply.
d. (x - 3) e. (x + 4)
Select the correct order of steps to rearrange the formula to solve for the acceleration.
d. Step 1: F = ma Step 2: F/m= ma/a Step 3: F/m= a
Is there a common factor? Factor the polynomial completely.
d. x(x - 22) (x + 25)
Is Adriana correct and why? a.) No, because there is more than one zero. b.) Yes, because 3 * 3 = 9 and 3 + 3 = 6 c.) Yes, because the two factors are (x-3)(x-3) d.) No, because the zero is at x = -3
d.) No, because the zero is at x = -3
Apply the order of operations to determine the correct factored from for the expression
e.
Completely factor
e.
Use quadratic techniques to factor the expression
e.
Select the factors that are part of the polynomial
e. a. c.
Complete the following statements about f( x ) or identify the following features of f( x ).
f(x) is an even function + ∞ + ∞
Apply the order of operations to determine the correct standard from for the expression
f.
link
link
What are the possible solutions for the equation?
x = +- 8
X = ? x = ?
x = -2 x = 4
X = ? x = ?
x = -2 x = 6
Which equations represent solutions or ways of finding the solutions to the equation 2x^2 + 7x -15 = 0
x = -7 +-_/169 ---------- 4 x = -5 (2x-3) (x+5) = 0
Select all exact solutions as well as decimal approximations to the hundredths.
x ≈ 2.39 and x ≈ 0.28 x = 4+_/10 ------ 3
Factor and make a conjunction.
x^2 − 16 = (x + 4) (x - 4) x^4 - 16 = (x^2 + 4) (x^2 - 4) Both of these equations are perfect squares and can use the perfect square formula to be factored. The only difference is that "x" is not a perfect square.
Factor and make a conjecture.
x^2- 18x + 81 = (x - 9) (x - 9) x^4- 18x^2+ 81 = (x^2- 9) (x^2 - 9) The only difference between these expressions is the 1st term. They both can be factored as a difference of squares.
Use long division to find the quotient
x^3 + x^2 -12x - 12 + 24 / x-1