Polynomial Division Practice Problems

To review these concepts, go to Polynomial Division Help and Synthetic Division Help

Polynomial Division Practice Problems

Problems

For Problems 1-4 use long division to find the quotient and remainder. For Problems 5 and 6, use synthetic division

1. (6 x ³ − 2 x ² + 5 x − 1) ÷ ( x ² + 3 x + 2)

2. ( x ³ − x ² + 2 x + 5) ÷ (3 x − 4)

6. ( x ³ + 8) ÷ ( x + 2)

7. Use synthetic division and the Remainder Theorem to evaluate f(c) .

   f(x) = 6 x ⁴ − 8 x ³ + x ² + 2 x − 5; c = −2

8. Completely factor the polynomial. f(x) = x ³ + 2 x ² − x − 2; c = 1 is a zero.

9. Completely factor the polynomial. P(x) = x ³ − 5 x ² + 5 x + 3; c = 3 is a zero.

Solutions

1.  

   

   The quotient is 6 x − 20, and the remainder is 53 x + 39.

2.  

   

3.  

   

    

   

   The quotient is −6 x + 2, and the remainder is 10 x .

4.  

   

   The quotient is x ² + x + 1, and the remainder is 0.

5.  

   

   The quotient is x ² − x + 4, and the remainder is −20.

6.  

   

   The quotient is x ² − 2 x + 4, and the remainder is 0.

7.  

   

   The remainder is 155, so f (−2) = 155.

8.  

   

   f(x) = ( x − 1)( x ² + 3 x + 2)

   = ( x − 1)( x + 1)( x + 2)

9.  

   

   P(x) = ( x − 3)( x ² − 2 x − 1)

   In order to factor x ² − 2 x − 1, we must first find its zeros.

   

   Because is a zero, = is a factor.

   Because is a zero, = is a factor.