Question

What must be subtracted from the polynomial f(x) = x⁴ + 2x³ - 13x² - 12x + 21 so that the resulting polynomial is exactly divisible by x² - 4x + 3?

Answer 1

Given:

the polynomial f(x) = x⁴ + 2x³ - 13x² - 12x + 21

To find:

What must be subtracted from the polynomial x⁴ + 2x³ - 13x² - 12x + 21 so that the resulting polynomial is exactly divisible by x² - 4x + 3

Solution:

Let g(x) be x² - 4x + 3

By applying division algorithm:

Dividend = Quotient× Divisor + Remainder

Dividend - Remainder = Quotient× Divisor

Now find the remainder when f(x)  is divisible by g(x).

r(x) = 2x - 3

Therefore,

r(x) = 2x - 3 has to be subtracted so that resulting polynomial is divisible by g(x)