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 Polynomials Polynomial Vocabulary Simplifying expressions Addition Subtraction Multiplication (FOIL) Multiplication Special products Long division Synthetic division Remainder theorem Roots and factors of a polynomial Polynomial Function Remainder theorem Remainder theorem. If p(x) is divides by (x-c) the remainder is a constant and and is equal to p(c) Supose that we divide the polynomial P(x)=2x3-5x+3 by x=1. Then according to the remainder theorem, the remainder in this case should be the number p(1). Let's check: The remainder is 0 and P(1) = 0. As the calculations show, the remainder is indeed equal to p(1) Use the remainder theorem to find the remainder when P(x)=2x3-5x+3 is divided by x-1. P(1)=2·13-5·1+3=2-5+3=0. Thus the remainder is 0. Proof of the remainder theorem To prove the remainder theorem we must show that when the polynomial P(x) is divided by x-a, the remainder is P(a). Now, according to the division algorithm, we can write P(x)=(x-a)·q(x) + r(x) (1) for unique polynomials q(x) and r(x). In this identity, either r(x) is the zero polynomial or the degree of r(x) is less than that of x-a. Since the degree of x-a is 1, the degree of r(x) must be zero. Thus in either case, the remainder r(x) is a constant. Denoting this constant by r, we can rewrite equation (1) as:P(x)=(x-a)·C(x) + r. If we set x=a in this identity we obtain P(a)=(a-a)·C(a) + r =r . We have now shown that P(a) = r. But by definition, r is the remainder r(x). Thus P(a) is the remainder. This proves the remainder theorem. Use the remainder theorem to evaluate the polynomial $\fs2P(x)=5x^4-32x^3-18x^2-26x+39$ when $\fs2x=7$ Solution: