Question

The expression x3 + ax2 – 15x + b has a factor x – 2 and leaves a remainder of 75 when divided by x + 3. Find the value of a and of b.

Answer 1

the polynomial remainder theorem states that the remainder of the division of a polynomial f(x) by a linear polynomial x-r is equal to f(r). In particular, x-r is a divisor (= a factor) of f(x), if and only if f(r)=0.

first we are dividing by x - 2.

so, r = 2.

as this is a factor, the remainder must be 0.

so,

0 = f(2) = 2³ + a2² - 15×2 + b = -22 + 4a + b

22 = 4a + b

then we divide by x + 3.

so, r = -3.

and the remainder is 75.

so,

75 = f(-3) = (-3)³ + a×(-3)² - 15×-3 + b = 18 + 9a + b

57 = 9a + b

now we have 2 equations with 2 variables to solve.

and since one term (+b) is the same in both equations, we use a little trick to subtract the first from the second equation.

57 = 9a + b

- 22 = 4a + b

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35 = 5a + 0

a = 7

22 = 4×7 + b = 28 + b

-6 = b

a = 7

b = -6