# B- Polynomials

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 What is the greatest common factor of the terms in the polynomial 3x³ – 3x² – 18x?▫️(FPC) 3x What is the greatest common factor of the terms in the polynomial 8x⁴ – 4x³ – 18x²? 2x² Which product of prime polynomials is equivalent to 8x⁴ + 36x³ – 72x²? 4x²(2x – 3)(x + 6) What is the greatest common factor of the terms in the polynomial 12x⁴ + 27x³ + 6x²? 3x² 5x³ + 20x² + 2x + 8 5x²(x + 4) + 2(x + 4) Which is the completely factored form of his polynomial? (5x² + 2) (x + 4) Which product of prime polynomials is equivalent to 8x³ – 2x? 2x(2x + 1)(2x – 1) 7x³ – 21x² + 5x – 15 7x²(x – 3) + 5(x -3) Which is the completely factored form of her polynomial? (7x² + 5) (x – 3) 12x⁴ + 30x³ + 4x² + 10x Ax(Bx² + 1) (2x + 5) B=[___________] 3 Which is the completely factored form of 4x³ + 10x² – 6x? 2x(x + 3)(2x – 1) Which is the completely factored form of 12x³ – 60x² + 4x – 20? 4(3x² + 1)(x – 5) 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 4x⁴ – 32x³ – 60x²? 4x² 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 12x⁴ + 27x³ + 6x²? 3x² Which polynomial is prime? 2x + 9 4x³ + 12x² + 5x + 15 4x²(x + 3) + 5(x + 3) Which is the completely factored form of her polynomial? (4×2 + 5)(x + 3) Which polynomial is prime? x² + 6 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 quotient (x³ + 6x² + 11x + 6) ÷ (x² + 4x + 3)? x + 2 The quotient of (x⁵ – 3x³ – 3x² – 10x + 15) and a polynomial is (x² – 5). What is the polynomial? x³ + 2x – 3 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 remainder of the following division problem? . .___1 x|x+1 . .x___ . .0+1 1 What is the first term of the quotient of the following division problem? (x³ – 1) ÷ (x + 2) x² What is the quotient (x³ + 8) ÷ (x + 2)? x² – 2x + 4 What is the first step of the following division problem? (8x³ – x² + 6x + 7) ÷ (2x – 1) Divide 8x³ by 2x. What is the quotient of (x³ + 3x² – 4x – 12) / (x² + 5x + 6)? [_________] x – 2 What is the quotient (x³ – 3x² + 5x – 3) ÷ (x – 1)? x² – 2x + 3 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 (3×4 – 4×2 + 8x – 1) ÷ (x – 2)? 3x³ + 6x² + 8x + 24 + 47/x-2 The quotient of (x⁴ + 5x³ – 3x – 15) and a polynomial is (x³ – 3). What is the polynomial? x + 5 What is the quotient (125 – 8x³) ÷ (25 + 10x + 4x²)? -2x + 5 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) x = -4, x = -2, or x = 7 _x = -7, x = 2, or x = 4 x = -7, x = 5, or x = 280 x = -280, x = -5, or x = 7 What remainder is represented by the synthetic division below? -5|2 10 1 5 . . |_-10 0-5 . . .2 .0 .1 0 0 Use synthetic division to solve (x³ + 1) ÷ (x – 1). What is the quotient? x³ – x² + x 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 – 4x² – 20x + 48 is x = 6. What are all the factors of the function? Use the Remainder Theorem. (x – 2)(x + 4)(x – 6) 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 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 Use synthetic division to solve (2x³ + 4x² – 35x + 15) / (x – 3). What is the quotient? 2x² + 10x -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 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 The volume of a rectangular prism is (x3 – 3×2 + 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 If (x + 8) is a factor of f(x), which of the following must be true? f(-8) = 0 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 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 If -1 is a root of f(x), which of the following must be true? A factor of f(x) is (x + 1). 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 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 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. What is the completely factored form of f(x) = x3 – 2×2 – 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) According to the Rational Root Theorem, which function has the same set of potential rational roots as the function g(x) = 3×5 – 2×4 + 9×3 – x2 + 12? f(x) = 3×5 – 2×4 – 9×3 + x2 – 12 According to the Rational Root Theorem, which could be a factor of the polynomial f(x) = 6×4 – 21×3 – 4×2 + 24x – 35? 2x – 7 According to the Rational Root Theorem, the following are potential roots of f(x) = 2×2 + 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 According to the Rational Root Theorem, which number is a potential root of f(x) = 9×8 + 9×6 – 12x + 7? 7/3 The graph of f(x) = x6 – 2×4 – 5×2 + 6 is shown below. How many roots of f(x) are rational numbers? (Roots: -183/250, -1, 1, 183/250) 2 The polynomial function f(x) = 10×6 + 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. According to the Rational Roots Theorem, which statement about f(x) = 25×7 – x6 – 5×4 + 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) = 66×4 – 2×3 + 11×2 + 35 is true? Any rational root of f(x) is a factor of 35 divided by a factor of 66. Using the Rational Root Theorem, what are all the rational roots of the polynomial f(x) = 20×4 + x3 + 8×2 + x – 12? -4/5 and 3/4 According to the Rational Root Theorem, what are all the potential rational roots of f(x) = 9×4 – 2×2 – 3x + 4? ±1/9, ±2/9, ±1/3, ±4/9, ±2/3, ±1, ±4/3, ±2, ±4 According to the Rational Root Theorem, what are all the potential rational roots of f(x) = 15×11 – 6×8 + x3 – 4x + 3? ±1/15, ±1/5, ±1/3, ±3/5, ±1, ±3 According to the Rational Root Theorem, -7/8 is a potential rational root of which function? f(x) = 24×7 + 3×6 + 4×3 – x – 28 According to the Rational Root Theorem, the following are potential roots of f(x) = 60x² – 57x – 18. -1/4 How many roots of f(x) are rational numbers? (Roots: 2, 3, 9/2) 3 What are the solutions of the equation x4 + 95×2 – 500 = 0? Use factoring to solve.▫️ x = ±√5 and x = ±10i Which equation is quadratic in form? _x6 + 6×4 + 8 = 0 7×6 + 36×3 + 5 = 0 4×9 + 20×3 + 25 = 0 Which quadratic equation is equivalent to (x2 – 1)2 – 11(x2 – 1) + 24 = 0? u2 – 11u + 24 = 0 where u = (x2 – 1) 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 What are the solutions of the equation x4 + 3×2 + 2 = 0? Use u substitution to solve. x = ±i√2 and x = ±i 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) What substitution should be used to rewrite 4×4 – 21×2 + 20 = 0 as a quadratic equation? u = x² Which equation is quadratic in form? _4(x – 2)2 + 3x – 2 + 1 = 0 8×5 + 4×3 + 1 = 0 10×8 + 7×4 + 1 = 0 9×16 + 6×4 + 1 = 0 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 – 9×3 + 8 = 0? Use u substitution to solve. x = 1 and x = 2 What are the solutions of the equation x4 – 5×2 – 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 substitution should be used to rewrite 4×12 – 5×6 – 14 = 0 as a quadratic equation? u = x⁶ Which quadratic equation is equivalent to (x – 4)2 – (x – 4) – 6 = 0? u2 – u – 6 = 0 where u = (x – 4) What substitution should be used to rewrite 6(x + 5)2 + 5(x + 5) – 4 = 0 as a quadratic equation? u = (x + 5) What are the solutions of the equation x6 + 6×3 + 5 = 0? Use factoring to solve. x = -³√5 and x = -1 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 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. Which statement describes the graph of f(x) = -x4 + 3×3 + 10×2? The graph touches the x axis at x = 0 and crosses the x axis at x = 5 and x = -2. Which of the following graphs could be the graph of the function f(x) = -0.08x(x² – 11x + 18)? . . . . \ . .| . . . . . \. | . ./\ _________\|__/____\_ . . . . . . . |. . . . . .| . . . . . . . |. . . . . .| . . . . . . . |. . . . . .| At which root does the graph of f(x) = (x – 5)³(x + 2)² touch the x axis? -2 What is the end behavior of the graph of the polynomial function f(x) = 3×6 + 30×5 + 75×4? 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 → ∞. At which root does the graph of f(x) = (x + 4)6(x + 7)5 cross the x axis? -7 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, ∞). Which of the following graphs could be the graph of the function 0.03x²(x² – 25)? . . . .|. . . .|. . . .| . . . .|. . . .|. . . .| ______|_____|_____|______ . . . . .\__/|\__/ . . . . . . . . | . . . . . . . . | 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 graph shows the end behavior of the graph of f(x) = 2x⁶ – 2x² – 5? . . . .|. . . . |. . . .| . . . . |. . . .|. . . | _______\__/|\__/_______ . . . . . . . . | . . . . . . . . | . . . . . . . . | Which graph has the same end behavior as the graph of f(x) = -3x³ – x² + 1? . .|. . . . . . .| . . | . . . . . | ____\______/|\___________ . . . .\__/. .|. .\ . . . . . . . . |. . .| . . . . . . . . |. . . | At which root does the graph of f(x) = (x – 5)³(x + 2)² touch the x axis? -2 Which of the following graphs could be the graph of the function f(x) = x⁴ + x³ – x² – x? . . . . . . .|. |. . .| . . . . . . .| .|. . .| ___________-|___/________ . . . . . . . . |.- . . . . . . . . |

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