2.7 Polynomials, the factor theorem and algebraic division — Learn

Prerequisites: expanding and collecting like terms; solving a quadratic by factorising from 2.4; the meaning of a factor.

By the end of this section you can: divide a polynomial by a linear expression and name the quotient and the remainder; use the remainder theorem to find a remainder without dividing; use the factor theorem to test whether a linear expression is a factor; and factorise a cubic completely.

1. Polynomials and their language

A polynomial is a sum of terms of the form $ax^n$, where $n$ is a whole number. The degree is the highest power of $x$.

is a polynomial of degree $3$. The number $2$ is the leading coefficient, and $-7$ is the constant term.

A polynomial of degree $n$ has at most $n$ roots, and at most $n$ linear factors. A cubic can have up to three roots, which is why factorising a cubic usually means finding one root and reducing the problem to a quadratic.

2. Algebraic division

Dividing a polynomial by a linear expression gives a quotient and a remainder. The relationship is always

where $Q(x)$ is the quotient, of degree one less than $p$, and $R$ is a constant remainder.

Example 1. Divide $x^3 + 2x^2 - 5x + 1$ by $x - 2$.

The quotient must be a quadratic, $x^2 + bx + c$. Look at the leading terms: $x \times x^2 = x^3$, which matches. Now work through the terms.

Collect the terms:

Compare with $x^3 + 2x^2 - 5x + 1$. The two agree except for the constant, which is $1$ instead of $-6$. So

The quotient is $x^2 + 4x + 3$ and the remainder is $7$.

3. The remainder theorem

To find the remainder when $p(x)$ is divided by $x - a$, substitute $x = a$. This is the remainder theorem:

Why? Put $x = a$ in $p(x) = (x - a)Q(x) + R$. The first term on the right vanishes, leaving $p(a) = R$.

Example 2. Use the remainder theorem to find the remainder when $x^3 + 2x^2 - 5x + 1$ is divided by $x - 2$.

Substitute $x = 2$:

Work out each term:

The remainder is $7$, matching the division in Example 1, with no division needed.

4. The factor theorem

A factor divides a polynomial exactly, leaving no remainder. The factor theorem is the remainder theorem with the remainder set to zero:

The symbol $\Longleftrightarrow$ means the two statements hold together: each one implies the other.

Example 3. Show that $x - 2$ is a factor of $p(x) = x^3 - 3x^2 + 4x - 4$.

Substitute $x = 2$:

The value is zero, so by the factor theorem $x - 2$ is a factor.

5. Factorising a cubic

To factorise a cubic, find one root, then divide to leave a quadratic. Factorise the quadratic in the usual way.

Example 4. Factorise $p(x) = x^3 - 6x^2 + 11x - 6$.

Try small whole numbers for $x$. The constant term is $-6$, so a whole-number root must be a factor of $6$: try $1$, $-1$, $2$, $-2$, $3$, $-3$.

So $x - 1$ is a factor. Divide to find the quotient:

Factorise the quadratic:

So

Example 5. Factorise $p(x) = x^3 + 2x^2 - x - 2$.

Again the constant term is $-2$, so try $1$ and $-1$.

so $x - 1$ is a factor. Divide:

Factorise the quadratic:

So

Watch out. The factor theorem tests one value, and the answer is a yes or a no.

A remainder of zero means the expression is a factor. A remainder that is not zero means it is not, and the division is not finished: write the remainder as well. If $p(2) = 7$, then $x - 2$ is not a factor, and $p(x) = (x - 2)Q(x) + 7$.

Re-check: multiply your quotient back by the divisor and add the remainder. You must recover the original polynomial.

Progress check

1. Work out the remainder when $x^3 - 4x^2 + 2x + 5$ is divided by $x - 3$.
2. Decide whether $x + 1$ is a factor of $x^3 - 7x + 6$.
3. Factorise $x^3 - 4x^2 + x + 6$ completely, and state its three roots.
4. A cubic has roots $1$, $2$ and $-3$. Write one possible cubic with these roots and a leading coefficient of $1$.

You should now be able to

• name the degree, the leading coefficient and the constant term of a polynomial;
• divide a polynomial by a linear expression and state the quotient and the remainder;
• use the remainder theorem to find a remainder by substitution;
• use the factor theorem to test a possible factor;
• factorise a cubic by finding one root and dividing.