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# Factor Theorem

### The Factor theorem

Question | Answer |
---|---|

Division Algorithm for Polynomials | If F(x) and G(x) Denote Polynomial functions and if g(x) is not the zero polynomial, then there are unique polynomial functions q(x) and r(x) such that where r(x) is either the zero polynomial or a polynomial of degree less than of g(x) |

Equation for Division Algorithm for Polynomials | F(x)/G(x) =q(x) + R(x)/G(x) or F(x) = G(x)q(x) + R(x) |

F(x) | Dividend |

G(x) | Divisor |

q(x) | quotient |

r(x) | Remainder |

Remainder theorem | If a polynomial f(x) is divided by (x - c), the remainder is f(c). |

Factor theorem | Then x - c is a factor of f(x) if and only if F(c) = 0 |

A factor of x³ + 16x² + 79x + 120 is A. x-5 B. x-3 C. x+3 D. x-8 | C |

For a polynomial P(x), if P(6) = 0, then which of the following must be a factor of P(x)? A. x+6 B. x-6 C. x²+6 D. x²-6 | B |

If -2x³ -6x² +5x-7 is divided by x-7 to give quotient of -2x²-20x-135 and remainder of -952, then which of following is true? A. (x-7)(-x²-20x-135) = -952 B. -2x³ -6x² +5x -7 = (x-7)(-x²-20x-135)-952 C. -2x³ -6x² +5x -7 = (x-7)(-x²-20x-135)+952 | B |

If P(x) = -8x³ -8x² -8x -8 is divided by x-3 what is the remainder? A. -160 B. -320 C. 320 D. 160 | B |

What is the remainder when x⁴ -8x² +9x +8 is divided by x+6? A. 1070 B. -1070 C. 962 D. -962 | C |

When P(x) = 5x³ -2x +2 is divided by 5x-2, the remainder is A. x² +x +12/5 B. P(2/5) = 38/25 C. P(5/2) = 601/8 D. P(-2) = -34 | B |

Which set of values for x should be tested to determine the possible zeroes of y = x³ +6x² -10x +35? A. ±5, ±7, and ±35 B. 1, 5, 7, and 35 C. 1, 5, 7, 12, and 35 D. ±1, ±5, ±7, and ±35 | D |

Based on the graph of f(x) = x⁴ -2x³ -24x² +8x +96, what are the real roots of x⁴ -2x³ -24x² +8x +96 =0? A. 6, 2, -2, -4 B. there are no real roots C. -6, -2, 2, 4 D. impossible to determine | A |

Determine the value of k so that x+2 is a factor of x³ +10x² +23x + k A. k = -14 B. k = -1 C. k = 1 D. k = 14 | D |

The fully factored form of x⁴ +10x³ +7x² -162x - 360 is A. (x+5)(x-3)(x-6)(x-4) B. x(x-5)(x-3)(x+24) C. (x-5)(x+3)(x+6)(x-6) D. (x+5)(x+3)(x+6)(x-4) | D |