B: Polynomials
What are the solutions of the equation x6 + 6x3 + 5 = 0? Use factoring to solve.
x = -³√5 and x = -1
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
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
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)
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
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, ∞).
What are the solutions of the equation x6 - 9x3 + 8 = 0? Use u substitution to solve.
x = 1 and x = 2
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 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
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 → ∞.
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.
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?
. . . .|. . . . |. . . .| . . . . |. . . .|. . . | _______\__/|\__/_______ . . . . . . . . | . . . . . . . . | . . . . . . . . |
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
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 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
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