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 \(3\):

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 \(-1\):