3.3 Dividing Polynomials
Ideas in this lesson
- Perform long division and state both division identities
- Divide by nonmonic linear divisors
- Divide by quadratic divisors with missing powers
- Perform synthetic division with signed or fractional inputs
- Evaluate with the remainder theorem and direct substitution
- Use a known zero to factor and find all other zeros
- Construct a polynomial from roots and a coefficient
- Construct integer-coefficient polynomials with irrational roots
- Recover polynomials from graphed roots and an additional point
- Use remainder and factor shortcuts for very high degrees
- Prove and use Horner form
- Prove the remainder theorem and both factor-theorem directions
Perform long division and state both division identities · Worked example
What are the quotient and remainder when \(3 x^{3} - x^{2} - x + 6\) is divided by \(x + 1\)?
Answer: \(Q(x)=3 x^{2} - 4 x + 3\); \(R=3\)
Arrange both polynomials in descending powers; retain a zero coefficient for each missing power.
Divide the leading terms to obtain \(3 x^{2}\).
Multiply the divisor: \(3 x^{3} + 3 x^{2}\). Subtract from \(3 x^{3} - x^{2} - x + 6\) to get \(- 4 x^{2} - x + 6\).
Divide the leading terms to obtain \(- 4 x\).
Multiply the divisor: \(- 4 x^{2} - 4 x\). Subtract from \(- 4 x^{2} - x + 6\) to get \(3 x + 6\).
Divide the leading terms to obtain \(3\).
Multiply the divisor: \(3 x + 3\). Subtract from \(3 x + 6\) to get \(3\).
The remainder is zero or has degree smaller than the divisor.
Check by expansion: \(3 x^{3} - x^{2} - x + 6=(x + 1)(3 x^{2} - 4 x + 3)+(3)\).
Divide the leading terms to obtain \(3 x^{2}\).
Multiply the divisor: \(3 x^{3} + 3 x^{2}\). Subtract from \(3 x^{3} - x^{2} - x + 6\) to get \(- 4 x^{2} - x + 6\).
Divide the leading terms to obtain \(- 4 x\).
Multiply the divisor: \(- 4 x^{2} - 4 x\). Subtract from \(- 4 x^{2} - x + 6\) to get \(3 x + 6\).
Divide the leading terms to obtain \(3\).
Multiply the divisor: \(3 x + 3\). Subtract from \(3 x + 6\) to get \(3\).
The remainder is zero or has degree smaller than the divisor.
Check by expansion: \(3 x^{3} - x^{2} - x + 6=(x + 1)(3 x^{2} - 4 x + 3)+(3)\).