Polynomials

What is the fifth term in the binomial expansion of (x + 5)⁸? A) 175,000x³ B) 43,750x⁴ C) 3,125x⁵ D) 7,000x⁵

B) 43,750x⁴

Which equation is quadratic in form? A) 6(x + 2)² + 8x + 2 + 1 = 0 B) 6x⁴ + 7x² - 3 = 0 C) 5x⁶ + x⁴ + 12 = 0 D) x⁹ + x³ - 10 = 0

B) 6x⁴ + 7x² - 3 = 0

What is the completely factored form of f(x) = 6x³ - 13x² - 4x + 15? A) (x + 1)(6x² - 19x + 15) B) (x + 1)²(2x - 3) C) (x + 1)(3x - 2)(5x - 3) D) (x + 1)(2x - 3)(3x - 5)

D) (x + 1)(2x - 3)(3x - 5)

According to the Rational Root Theorem, which is a factor of the polynomial f(x) = 3x³ - 5x² - 12x + 20? A) 2x + 1 B) 2x - 1 C) 3x + 5 D) 3x - 5

D) 3x - 5

Which polynomial function has a leading coefficient of 1, roots -2 and 7 with multiplicity 1, and root 5 with multiplicity 2? A) f(x) = 2(x + 7)(x + 5)(x - 2) B) f(x) = 2(x - 7)(x - 5)(x + 2) C) f(x) = (x + 7)(x + 5)(x + 5)(x - 2) D) f(x) = (x - 7)(x - 5)(x - 5)(x + 2)

D) f(x) = (x - 7)(x - 5)(x - 5)(x + 2)

Three roots of a fifth degree polynomial function f(x) are -2, 2, and 4 + i. Which statement describes the number and nature of all roots for this function? A) f(x) has two real roots and one imaginary root. B) f(x) has three real roots. C) f(x) has five real roots. D) f(x) has three real roots and two imaginary roots.

D) f(x) has three real roots and two imaginary roots.

Which of the following describes the roots of the polynomial function f(x)=(x+2)²(x-4)(x+1)³? A) -2 with multiplicity 2, 4 with multiplicity 1, and -1 with multiplicity 3 B) -2 with multiplicity 3, 4 with multiplicity 2, and -1 with multiplicity 4 C) 2 with multiplicity 2, -4 with multiplicity 1, and 1 with multiplicity 3 D) 2 with multiplicity 3, -4 with multiplicity 2, and 1 with multiplicity 4

A) -2 with multiplicity 2, 4 with multiplicity 1, and -1 with multiplicity 3

The price that a company charged for a basketball hoop is given by the equation 50-5x² where x is the number of hoops that are produced, in millions. It costs the company $30 to make each basketball hoop. The company recently reduced its production to 1 million hoops but maintained its profit of 15 million dollars. Approximately how many basketball hoops did the company previously produce to make the same profit? A) 1.3 million hoops B) 1.4 million hoops C) 15 million hoops D) 30 million hoops

A) 1.3 million hoops

The graphs of two polynomial equations do not intersect. Kelly concludes that the system has no solution. Which statement best explains why Kelly's reasoning is correct or incorrect? A) She is incorrect because systems may have only complex solutions which are not visible on a graph. B) She is incorrect because the solutions may have multiplicity greater than 1. C) She is correct because all solutions are intersection points of the graphs. D) She is correct because the solutions are irrational.

A) She is incorrect because systems may have only complex solutions which are not visible on a graph.

Which statement best describes the equation (x + 5)² + 4(x + 5) + 12 = 0? A) The equation is quadratic in form because it can be rewritten as a quadratic equation with u substitution u = (x + 5). B) The equation is quadratic in form because when it is expanded, it is a fourth-degree polynomial. C) The equation is not quadratic in form because it cannot be solved by using the quadratic formula. D) The equation is not quadratic in form because there is no real solution.

A) The equation is quadratic in form because it can be rewritten as a quadratic equation with u substitution u = (x + 5).

Which statement describes the graph of this polynomial function? f(x)=x⁵-6x⁴+9x³ A) The graph crosses the x-axis at x = 0 and touches the x-axis at x = 3. B) The graph touches the x-axis at x = 0 and crosses the x-axis at x = 3. C) The graph crosses the x-axis at x = 0 and touches the x-axis at x = -3. D) The graph touches the x-axis at x = 0 and crosses the x-axis at x = -3.

A) The graph crosses the x-axis at x = 0 and touches the x-axis at x = 3.

A polynomial function has a root of -6 with multiplicity 1, a root of -2 with multiplicity 3, a root of 0 with multiplicity 2, and a root of 4 with multiplicity 3. If the function has a positive leading coefficient and is of odd degree, which statement about the graph is true? A) The graph of the function is positive on (-6, -2). B) The graph of the function is negative on (∞, 0). C) The graph of the function is positive on (-2, 4). D) The graph of the function is negative on (4, ∞).

A) The graph of the function is positive on (-6, -2).

If (x - 2k) is a factor of f(x), which of the following must be true? A) f(2k) = 0 B) f(-2k) = 0 C) A root of f(x) is x = -2k. D) A y intercept of f(x) is x = 2k.

A) f(2k) = 0

Which expression represents the sixth term in the binomial expansion of (5y + 3)¹⁰? A) ₁₀C₅(5y)⁵(3)⁵ B) ₁₀C₆(5y)⁴(3)⁶ C) 5₁₀C₅(y)⁵(3)⁵ D) 5₁₀C₆(y)⁴(3)⁶

A) ₁₀C₅(5y)⁵(3)⁵

What are the roots of the polynomial equation x³-7x=6x-12? Use a graphing calculator and a system of equations. A) -6, 6 B) -4, 1, 3 C) -3, -1, 4 D) 1, 3

B) -4, 1, 3

A cube has side length x. One side of the cube is increased by 4 inches, and another side is doubled. The volume of the new rectangular prism is 450 cubic inches. The equation 2x²+8x²=450 can be used to find x. What was the side length of the original cube? Use a graphing calculator and a system of equations to find the answer. A) 4 inches B) 5 inches C) 9 inches D) 10 inches

B) 5 inches

Which polynomial function has a leading coefficient of 2, root -4 with multiplicity 3, and root 10 with multiplicity 1? A) f(x) = 2(x - 4)(x - 4)(x - 4)(x + 10) B) f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10) C) f(x) = 3(x - 4)(x - 4)(x + 10) D) f(x) = 3(x + 4)(x + 4)(x - 10)

B) f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10)

What are the solutions of the equation (x - 3)² + 2(x - 3) - 8 = 0? Use u substitution to solve. A) x = -5 and x = 1 B) x = -1 and x = 5 C) x = -1 and x = -7 D) x = 1 and x = 7

B) x = -1 and x = 5

If f(-2) = 0, what are all the factors of the function f(x)=x³-2x²-68x-120? Use the Remainder Theorem. A) (x + 2)(x + 60) B) (x - 2)(x - 60) C) (x - 10)(x + 2)(x + 6) D) (x + 10)(x - 2)(x - 6)

C) (x - 10)(x + 2)(x + 6)

What is the completely factored form of x² - 16xy + 64y²? A) xy(x - 16 + 64y) B) xy(x + 16 + 64y) C) (x - 8y)(x - 8y) D) (x + 8y)(x + 8y)

C) (x - 8y)(x - 8y)

What is the remainder when (x⁴ + 36) is divided by (x² - 8)? A) 36 B) 64 C) 100 D) 117

C) 100

Which statement about -2h² - 15h - 7 is true? A) One of the factors is (h + 2). B) One of the factors is (3h - 2). C) One of the factors is (2h + 1). D) One of the factors is (h - 7).

C) One of the factors is (2h + 1).

What is the polynomial function of lowest degree with lead coefficient 1 and roots 1 and 1 + i? A) f(x) = x² - 2x + 2 B) f(x) = x³ - x² + 4x - 2 C) f(x) = x³ - 3x² + 4x - 2 D) f(x) = x² - x + 2

C) f(x) = x³ - 3x² + 4x - 2

Use synthetic division to solve (x4 - 1) ÷ (x - 1). What is the quotient? A) x³-x²+x-1 B) x³ C) x³+x²+x+1 D) x³ - 2

C) x³+x²+x+1