4.3 – The Remainder and Factor Theorems

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Remainder and Factor Theorems

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Presentation transcript:

4.3 – The Remainder and Factor Theorems

“I can use the Remainder and Factor Theorems to find factors of polynomials” REMAINDER THEOREM: If a polynomial P(x) is divided by x – r, the remainder is a constant P(r), and P(x) = (x – r) ∙ Q(x) + P(r), where Q(x) is a polynomial with degree one less than the degree of P(x).

“I can use the Remainder and Factor Theorems to find factors of polynomials” REMAINDER THEOREM: Example: Let P(x) = 2x2 + 3x – 8, first divide P(x) by x – 2. Next, find P(2) First Answer: 2a + 7 with a remainder of 6 Second Answer: P(2) = 6

“I can use the Remainder and Factor Theorems to find factors of polynomials” FACTOR THEOREM: The binomial x – r is a factor of the polynomial P(x) if and only if P(r) = 0. Example: Determine if x – 1 is a factor of P(x) = 2x3 – 3x2 + x. Answer: yes, because P(1) = 0

“I can divide polynomials using synthetic division.” A short-cut to long division. Only works when dividing by a binomial of the form x – r. Example: Divide –y6 + 4y4 + 3y2 + 2y by y + 2. Answer: -y5 + 2y4 + 3y – 4 remainder 8 *Note: substitute zero place holders since all exponents are not accounted for in the problem!

“I can factor polynomials using the Factor Theorem and synthetic division.” RATIONAL ROOTS THEOREM: Let a0xn + a1xn-1 + … + an-1x + an = 0 represent a polynomial equation of degree n with integral coefficients. If a rational number p/q, where p and q have no common factors, is a root of the equation, then p is a factor of an and q is a factor of a0. Example: All possible rational roots of 6x3 + 11x2 – 3x – 2 = 0 are p = +1, +2 q = +1, +2. +3, + 6  p/q = +1, +1/2, +1/3, +1/6, +2, and +2/3

Determine the binomial factors of x3 – 2x2 – 13x – 10. First, find a rational root by using the Rational Root and Factor Theorems. Second, use synthetic division with the rational root you found in the first part. Third, repeat steps 1 and 2 until you have a quadratic equation left to factor.

Determine the binomial factors of x3 – 2x2 – 13x – 10. First, find a rational root by using the Rational Root and Factor Theorems. All possible roots: (all pos and neg) 1, 2, 5, 10. -1, -2, and 5 work

Determine the binomial factors of x3 – 2x2 – 13x – 10 Second, use synthetic division with the rational root you found in the first part. Just divide 1 time with the first root you found from -1, -2, and 5.

Determine the binomial factors of x3 – 2x2 – 13x – 10 Third, repeat steps 1 and 2 until you have a quadratic equation left to factor. Final answer (x + 1)(x + 2)(x – 5)

Homework 4.3 p226 #29 – 34, 37, 38, 49