2.1, 3.1 - Dividing polynomials

Polynomial division

A polynomial is a finite sum of terms in non-negative integer powers of $x$. Dividing a polynomial $f(x)$ by a nonzero polynomial $d(x)$ means finding a quotient $q(x)$ and remainder $r(x)$ such that

The remainder is zero or has degree less than the divisor. The degree is the largest power with a nonzero coefficient.

Worked example 1: a nonzero remainder

Question. Find the quotient and remainder when $x^2+3x+5$ is divided by $x+1$.

Plan. Match the squared term first, then express the dividend as a multiple of the divisor plus a constant.

Answer. Quotient $x+2$, remainder $3$.

Check. Multiplying $(x+1)(x+2)$ gives $x^2+3x+2$; adding $3$ recovers the dividend.

Worked example 2: exact division

Question. Find the quotient and remainder when $x^2+3x+2$ is divided by $x+1$.

Plan. Match the same leading terms and examine what is left.

Answer. Quotient $x+2$, remainder $0$. The divisor is a factor, since it divides exactly.

Check. The product expands to all three terms of the dividend, with no correction needed.

Watch out

The remainder is $3$, not $3/(x+1)$. The latter is the extra fraction in the division result; the polynomial identity uses $+3$.

Worked example 3: a missing linear term

Question. Divide $2x^3+3x^2+5$ by $x+1$.

Plan. Cancel the cubic, quadratic and linear terms in turn, subtracting a complete multiple each time.

The first quotient term is $2x^3/x=2x^2$:

Next use $x^2/x=x$:

Finally use $-x/x=-1$:

Answer. Quotient $2x^2+x-1$, remainder $6$.

Check. $(x+1)(2x^2+x-1)+6=2x^3+3x^2+0x+5$.