Polynomial division is used to divide one polynomial by another, usually by a linear factor such as \(x-a\).
When dividing polynomials, we write
\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]
If the divisor is linear, the remainder is a constant.
If \(f(x)\) is divided by \(x-a\), then
\[ f(x)=(x-a)Q(x)+R \]
where \(Q(x)\) is the quotient and \(R\) is the remainder.
Example: divide \(2x^3+3x^2-5x+6\) by \(x+2\).
Step 1: Divide the first term:
\[ 2x^3 \div x = 2x^2 \]
Put \(2x^2\) in the quotient.
Step 2: Multiply back:
\[ 2x^2(x+2)=2x^3+4x^2 \]
Step 3: Subtract:
\[ (2x^3+3x^2-5x+6)-(2x^3+4x^2)=-x^2-5x+6 \]
Step 4: Repeat:
\[ -x^2 \div x = -x \]
Put \(-x\) in the quotient, then multiply back:
\[ -x(x+2)=-x^2-2x \]
Subtract:
\[ (-x^2-5x+6)-(-x^2-2x)=-3x+6 \]
Repeat again:
\[ -3x \div x = -3 \]
Multiply back:
\[ -3(x+2)=-3x-6 \]
Subtract:
\[ (-3x+6)-(-3x-6)=12 \]
Always include missing powers of \(x\).
Example: divide \(x^3-5x+2\) by \(x-1\).
Write the dividend as
\[ x^3+0x^2-5x+2 \]
This makes the division clearer and avoids errors.
Synthetic division is a quicker method when dividing by a linear factor of the form \(x-a\).
Example: divide \(x^3-2x^2-5x+6\) by \(x-3\).
Use the coefficients \(1,-2,-5,6\) and the number \(3\):
| 3 | 1 | -2 | -5 | 6 |
| 3 | 3 | -6 | ||
| 1 | 1 | -2 | 0 |
So the quotient is
\[ x^2+x-2 \]
and the remainder is \(0\).
If \(f(x)\) is divided by \(x-a\), then the remainder is \(f(a)\).
Example:
\[ f(x)=x^3-2x^2-5x+6 \]
When dividing by \(x-3\), the remainder is
\[ f(3)=27-18-15+6=0 \]
So \(x-3\) is a factor.
Divide \(x^3+4x^2-x+7\) by \(x+1\).
Using synthetic division with \(-1\):
| -1 | 1 | 4 | -1 | 7 |
| -1 | -3 | 4 | ||
| 1 | 3 | -4 | 11 |