Dividing Polynomials

If you divide a polynomial by (x - a), instead of doing long division, just substitute x = a into the original polynomial. The value obtained is your remainder.

(1) Find the remainder when p(x) = x³ - 2x + 1 is divided by (x - 2).

To use remainder theorem, we put x = into p(x).

So, p(2) = (2)³ - 2(2) + 1 = - + =

Therefore, remainder = 5.

(2) Find the remainder when p(x) = 2x⁴ - 3x² + 4x - 1 is divided by (x + 3).

For (x + 3), we put x = into p(x)

p(-3) = 2(-3)⁴ - 3(-3)² + 4(-3) - 1 = 2 × - 3 × + 4 × - 1 = + + + =

Therefore, remainder = 122.

(3) Find the remainder when p(x) = x³ + x² - 2x + 1 is divided by (x - 1).

We have to put x = into p(x).

p(1) = (1)³ + (1)² - 2(1) + 1 = + - + =

Therefore, remainder = 1

(4) Use the remainder theorem to check if (x - 2) is a factor of p(x) = x³ - 3x² + 2x + 4

If (x - 2) is a factor, then p(2)p(-2) should be .

p(2) = (2)³ - 3(2)² + 2(2) + 4 = + + + =

Since p(2) ≠ 0, (x - 2) is notis a factor of p(x).