Polynomial factoring/division

Question

1. Suppose f and g are polynomials and that f(2)=4 and g(2)= -2. Find the remainder of each of the following when you divide each polynomial by x −2. a. 2f(x)+g(x)

b. f^2(x)+(x-3)g(x)

2. Consider f(x)= x^3+10x^2+31x+30 and complete each of the following:

a. Find f(2), f(1), f(0), f(-1), & f(-2)

b. Divide f(x) by a factor of f. Can use part a for a hint.

c. Completely factor f(x) into a product of linear factors. Ex: (x-1)(x-2)(x-3)

3. Suppose P(x)= 4x^3+x^2-3x+1. Use the Remainder Theorem to show that P(x)-P(2)is divisable by x-2.

Answer 1

Remainder Theorem:

When dividing a polynomial f(x) by (x-c), then remainder is f(c).

Factor Theorem:

When f(c) = 0 then (x-c) is a factor of f(x)

1) Remainder of:

a) 2f(x) + g(x) = 2f(2) + g(2) = 2(4)-2 = 6

b) f²(x) + (x-3)g(x) = 16 + (-1)(-2) = 18

2) f(x) = x³+10x²+31x+30

a)

f(2) = 8+40+62+30 = 140

f(1) = 1+10+31+30= 72

f(0) = 30

f(-1) = -1+10-31+30 = 8

f(-2) = -8 +40 -62+30 = 0

b) -2 is a factor of f(x) because the remainder is 0

By synthetic division

1 10 31 30 |-2

-2 -16 -30

------------------------

1 8 15 0

f(x)/(x+2) = x²+8x+15

c) = (x+2)(x²+8x+15) = (x+2)(x+3)(x+5)

3) P(x) = 4x³+x²-3x + 1

P(2) = 32+4-6+1 = 31

g(x) = P(x)-P(2) = 4x³+x²-3x -30

g(2) = 32+4-6-30 = 0

Since the remainder is zero, (x-2) is a factor of P(x) - P(2)