B: Polynomials

Which graph has the same end behavior as the graph of f(x) = -3x³ - x² + 1?

. .|. . . . . . .| . . | . . . . . | ____\______/|\___________ . . . .\__/. .|. .\ . . . . . . . . |. . .| . . . . . . . . |. . . |

Which equation is quadratic in form?

_x6 + 6x4 + 8 = 0 7x6 + 36x3 + 5 = 0 4x9 + 20x3 + 25 = 0

Which quadratic equation is equivalent to (x + 2)2 + 5(x + 2) - 6 = 0?

(u + 2)2 + 5(u + 2) - 6 = 0 where u = (x - 2) u2 + 4 + 5u - 6 = 0 where u = (x - 2) _u2 + 5u - 6 = 0 where u = (x + 2) u2 + u - 6 = 0 where u = (x + 2)

3x⁴ - 18x³ + 9x² -54x Ax(x² + B) (x + C) C=[___________]

-6

Which is the completely factored form of 3x² - 12x - 15?

3(x + 1)(x - 5)

What is the greatest common factor of the terms in the polynomial 4x⁴ - 32x³ - 60x²?

4x²

If (x + 8) is a factor of f(x), which of the following must be true?

f(-8) = 0

What is the completely factored form of f(x) = x3 - 2x2 - 5x + 6?

f(x) = (x + 2)(x - 3)(x + 6) f(x) = (x + 2)(x - 3)(x - 6) f(x) = (x - 2)(x + 3)(x - 1) _f(x) = (x + 2)(x - 3)(x - 1)

4x³ + 12x² + 5x + 15 4x²(x + 3) + 5(x + 3) Which is the completely factored form of her polynomial?

(4x2 + 5)(x + 3)

5x³ + 20x² + 2x + 8 5x²(x + 4) + 2(x + 4) Which is the completely factored form of his polynomial?

(5x² + 2) (x + 4)

7x³ - 21x² + 5x - 15 7x²(x - 3) + 5(x -3) Which is the completely factored form of her polynomial?

(7x² + 5) (x - 3)

According to the Rational Root Theorem, the following are potential roots of f(x) = 60x² - 57x - 18.

-1/4

At which root does the graph of f(x) = (x - 5)³(x + 2)² touch the x axis?

-2

What is the quotient (125 - 8x³) ÷ (25 + 10x + 4x²)?

-2x + 5

Using the Rational Root Theorem, what are all the rational roots of the polynomial f(x) = 20x4 + x3 + 8x2 + x - 12?

-4/5 and 3/4

The graph of f(x) = x6 - 2x4 - 5x2 + 6 is shown below. How many roots of f(x) are rational numbers? (Roots: -183/250, -1, 1, 183/250)

2

The area of a rectangle is 5x³ + 19x² + 6x - 18 with length x + 3. Using synthetic division, what is the width of the rectangle?

5x² + 4x - 6

According to the Rational Root Theorem, which number is a potential root of f(x) = 9x8 + 9x6 - 12x + 7?

7/3

What is the end behavior of the graph of the polynomial function f(x) = 3x6 + 30x5 + 75x4?

As x → -∞, y → -∞ and as x → ∞, y → -∞. As x → -∞, y → -∞ and as x → ∞, y → ∞. As x → -∞, y → ∞ and as x → ∞, y → -∞. _As x → -∞, y → ∞ and as x → ∞, y → ∞.

A polynomial function has a root of -5 with multiplicity 3, a root of 1 with multiplicity 2, and a root of 3 with multiplicity 7. If the function has a negative leading coefficient and is of even degree, which statement about the graph is true?

The graph of the function is negative on (3, ∞).

According to the Rational Root Theorem, the following are potential roots of f(x) = 2x2 + 2x - 24. -4, -3, 2, 3, 4 Which are actual roots of f(x)?

_-4 and 3 -4, 2, and 3 -3 and 4 -3, 2, and 4

Which equation is quadratic in form?

_4(x - 2)2 + 3x - 2 + 1 = 0 8x5 + 4x3 + 1 = 0 10x8 + 7x4 + 1 = 0 9x16 + 6x4 + 1 = 0

What is the quotient (x³ + 6x² + 11x + 6) ÷ (x² + 4x + 3)?

x + 2

The long division below shows the first term of the quotient. Which polynomial should be subtracted from the dividend first? . . . .____x²____ x+2|x³+3x²+x▫️(DoP)

x³ + 2x²

What is the first step of the following division problem? (8x³ - x² + 6x + 7) ÷ (2x - 1)

Divide 8x³ by 2x.

What are the solutions of the equation x4 + 95x2 - 500 = 0? Use factoring to solve.▫️

x = ±√5 and x = ±10i

The volume of a rectangular prism is (x3 - 3x2 + 5x - 3), and the area of its base is (x2 - 2). If the volume of a rectangular prism is the product of its base area and height, what is the height of the prism?

x-3 + 7x-9/x²-2

What is the quotient of the following division problem? . . . ._____x+2 x+1|x²+3x+2 . . . .____x²+x . . . .0+2x+2 . . . .___2x+2 . . . . . . . . 0

x² + 3x + 2

One root of f(x) = x³ - 4x² - 20x + 48 is x = 6. What are all the factors of the function? Use the Remainder Theorem.

(x - 2)(x + 4)(x - 6)

At which root does the graph of f(x) = (x + 4)6(x + 7)5 cross the x axis?

-7

Which of the following graphs could be the graph of the function f(x) = x⁴ + x³ - x² - x?

. . . . . . .|. |. . .| . . . . . . .| .|. . .| ___________-|___/________ . . . . . . . . |.- . . . . . . . . |

Which of the following graphs could be the graph of the function f(x) = -0.08x(x² - 11x + 18)?

. . . . \ . .| . . . . . \. | . ./\ _________\|__/____\_ . . . . . . . |. . . . . .| . . . . . . . |. . . . . .| . . . . . . . |. . . . . .|

Which graph shows the end behavior of the graph of f(x) = 2x⁶ - 2x² - 5?

. . . .|. . . . |. . . .| . . . . |. . . .|. . . | _______\__/|\__/_______ . . . . . . . . | . . . . . . . . | . . . . . . . . |

Which of the following graphs could be the graph of the function 0.03x²(x² - 25)?

. . . .|. . . .|. . . .| . . . .|. . . .|. . . .| ______|_____|_____|______ . . . . .\__/|\__/ . . . . . . . . | . . . . . . . . |

One factor of f(x) = 5x³ + 5x² - 170x + 280 is (x + 7). What are all the roots of the function? Use the Remainder Theorem.▫️(SD&tRT)

...

What remainder is represented by the synthetic division below? -5|2 10 1 5 . . |_-10 0-5 . . .2 .0 .1 0

0

What is the remainder of the following division problem? . .___1 x|x+1 . .x___ . .0+1

1

What is the greatest common factor of the terms in the polynomial 3x³ - 3x² - 18x?▫️(FPC)

3x

Which product of prime polynomials is equivalent to 3x⁴ - 81x?

3x(x - 3)(x² + 3x + 9)

What is the greatest common factor of the terms in the polynomial 12x⁴ + 27x³ + 6x²?

3x²

Use synthetic division to solve (3x⁴ + 6x³ + 2x² + 9x + 10) / (x - 2). What is the quotient?

3x³ + 12x² + 26x + 61 + 132/x + 2 _3x³ + 2x + 5 3x³ + 12x² + 26x + 61 + 132/x - 2 3x⁴ + 2x² + 5x

What is the quotient (3x4 - 4x2 + 8x - 1) ÷ (x - 2)?

3x³ + 6x² + 8x + 24 + 47/x-2

Which is the completely factored form of 12x³ - 60x² + 4x - 20?

4(3x² + 1)(x - 5)

Use synthetic division to solve (4x³ - 3x² + 5x + 6) / (x + 6). What is the quotient?

4x² - 27x + 167 - 996/x - 6 4x² + 21x + 131 + 792/x + 6 4x² + 21x + 131 + 792/x - 6 _4x² - 27x + 167 - 996/x + 6

Which product of prime polynomials is equivalent to 8x⁴ + 36x³ - 72x²?

4x²(2x - 3)(x + 6)

What divisor is represented by the synthetic division below? -5|2 10 1 5 . . |_-10 0-5 . . .2 .0 .1 0

5 _x + 5 x - 5

If -1 is a root of f(x), which of the following must be true?

A factor of f(x) is (x + 1).

According to the Rational Roots Theorem, which statement about f(x) = 25x7 - x6 - 5x4 + x - 49 is true?

Any rational root of f(x) is a factor of -49 divided by a factor of 25.

According to the Rational Root Theorem, which statement about f(x) = 66x4 - 2x3 + 11x2 + 35 is true?

Any rational root of f(x) is a factor of 35 divided by a factor of 66.

According to the Rational Root Theorem, which statement about f(x) = 12x³ - 5x² + 6x + 9?▫️

Any rational root of f(x) is a multiple of 12 divided by a multiple of 9. Any rational root of f(x) is a multiple of 9 divided by a multiple of 12. _Any rational root of f(x) is a factor of 9 divided by a factor of 12.

Which statement describes the graph of f(x) = 4x⁷ + 40x⁶ + 100x⁵?

The graph crosses the x axis at x = 0 and touches the x axis at x = -5.

What is the solution of the equation (x - 5)2 + 3(x - 5) + 9 = 0? Use u substitution and the quadratic formula to solve.

x = -3±3i√3/2 _x = 7±3i√3/2 x = 2 x = 8

What are the solutions of the equation x6 + 6x3 + 5 = 0? Use factoring to solve.

x = -³√5 and x = -1

What are the solutions of the equation x6 - 9x3 + 8 = 0? Use u substitution to solve.

x = 1 and x = 2

One factor of f(x) = 4x³ - 4x² - 16x + 16 is (x - 2). What are all the roots of the function? Use the Remainder Theorem.

x = 1, x = 2, or x = 4 _x = -2, x = 1, or x = 2 x = 2, x = 4, or x = 16 x = -16, x = 2, or x = 16

One root of f(x) = x³ - 9x² + 26x - 24 is x = 2. What are all the roots of the function? Use the Remainder Theorem.

x = 2, x = 3, or x = 4

What are the solutions of the equation x4 - 5x2 - 14 = 0? Use factoring to solve.

x = ± √7 and x = ± √2 x = ±i√7 and x = ±i√2 x = ±i√7 and x = ± √2 _x = ± √7 and x = ±i√2

What are the solutions of the equation x4 + 3x2 + 2 = 0? Use u substitution to solve.

x = ±i√2 and x = ±i

What is the first term of the quotient of the following division problem? (x³ - 1) ÷ (x + 2)

x²

Which polynomial is prime?

x² + 6

What is the quotient (x³ - 3x² + 5x - 3) ÷ (x - 1)?

x² - 2x + 3

What is the quotient (x³ + 8) ÷ (x + 2)?

x² - 2x + 4

The quotient of (x⁵ - 3x³ - 3x² - 10x + 15) and a polynomial is (x² - 5). What is the polynomial?

x³ + 2x - 3

Use synthetic division to solve (x³ + 1) ÷ (x - 1). What is the quotient?

x³ - x² + x

According to the Rational Root Theorem, what are all the potential rational roots of f(x) = 15x11 - 6x8 + x3 - 4x + 3?

±1/15, ±1/5, ±1/3, ±3/5, ±1, ±3

According to the Rational Root Theorem, what are all the potential rational roots of f(x) = 9x4 - 2x2 - 3x + 4?

±1/9, ±2/9, ±1/3, ±4/9, ±2/3, ±1, ±4/3, ±2, ±4

Which polynomial is prime?

2x + 9

According to the Rational Root Theorem, which could be a factor of the polynomial f(x) = 6x4 - 21x3 - 4x2 + 24x - 35?

2x - 7

Which product of prime polynomials is equivalent to 8x³ - 2x?

2x(2x + 1)(2x - 1)

Which is the completely factored form of 4x³ + 10x² - 6x?

2x(x + 3)(2x - 1)

What is the greatest common factor of the terms in the polynomial 8x⁴ - 4x³ - 18x²?

2x²

Use synthetic division to solve (2x³ + 4x² - 35x + 15) / (x - 3). What is the quotient?

2x² + 10x -5

The volume of a rectangular prism is 2x³ + 9x² - 8x - 36 with height x + 2. Using synthetic division, what is the area of the base?

2x² + 5x - 18

What dividend is represented by the synthetic division below? -5|2 10 1 5 . . |_-10 0-5 . . .2 .0 .1 0

2x³ + 10x² + x + 5

12x⁴ + 30x³ + 4x² + 10x Ax(Bx² + 1) (2x + 5) B=[___________]

3

How many roots of f(x) are rational numbers? (Roots: 2, 3, 9/2)

3

Which statement describes the graph of f(x) = -4x³ - 28x² - 32x + 64?

The graph touches the x axis at x = -4 and crosses the x axis at x = 1.

Which statement describes the graph of f(x) = -x4 + 3x3 + 10x2?

The graph touches the x axis at x = 0 and crosses the x axis at x = 5 and x = -2.

The polynomial function f(x) = 10x6 + 7x - 7 is graphed below. According to the rational roots theorem, which is a possible root at point P?

The root at point P may be 7/10.

Let a and b be real numbers where a ≠ b ≠ 0. Which of the following functions could represent the graph below? . .\. . | . / ____\/|\/____ . . . . .| . . . . .|▫️

f(x) = (x - a)²(x - b)⁴

According to the Rational Root Theorem, -7/8 is a potential rational root of which function?

f(x) = 24x7 + 3x6 + 4x3 - x - 28

According to the Rational Root Theorem, which function has the same set of potential rational roots as the function g(x) = 3x5 - 2x4 + 9x3 - x2 + 12?

f(x) = 3x5 - 2x4 - 9x3 + x2 - 12

What substitution should be used to rewrite 6(x + 5)2 + 5(x + 5) - 4 = 0 as a quadratic equation?

u = (x + 5)

What substitution should be used to rewrite 4x4 - 21x2 + 20 = 0 as a quadratic equation?

u = x²

What substitution should be used to rewrite 4x12 - 5x6 - 14 = 0 as a quadratic equation?

u = x⁶

Which quadratic equation is equivalent to (x2 - 1)2 - 11(x2 - 1) + 24 = 0?

u2 - 11u + 24 = 0 where u = (x2 - 1)

Which quadratic equation is equivalent to (x - 4)2 - (x - 4) - 6 = 0?

u2 - u - 6 = 0 where u = (x - 4)

The quotient of (x⁴ + 5x³ - 3x - 15) and a polynomial is (x³ - 3). What is the polynomial?

x + 5

If (x2 - 4) ÷ (x + 2) = x - 2, which polynomial should fill in the blank below? (x + 2) ´ ______ = x2 - 4

x - 2

What is the quotient of (x³ + 3x² - 4x - 12) / (x² + 5x + 6)? [_________]

x - 2

What are the solutions of the quadratic equation (x - 8)2 - 13(x - 8) + 30 = 0? Use u substitution to solve.

x = -11 and x = -18 x = -2 and x = 5 x = 2 and x = -5 _x = 11 and x = 18

If f(1) = 0, what are all the roots of the function x³ + 3x² - x - 3? Use the Remainder Theorem.

x = -3, x = -1, or x = 1