TMUA Polynomials and the Factor Theorem

Remainder theorem

When $p(x)$ is divided by $(x-a)$, the remainder is $p(a)$. When divided by $(x-a)(x-b)$, the remainder has the form $cx+d$.

Factor theorem

$(x-a)$ is a factor of $p(x)$ exactly when $p(a)=0$. Use this to find a missing coefficient, then divide to find the other factors.

Algebraic division

Divide term by term, or compare coefficients after writing $p(x)=(x-a)q(x)+r$.

Roots, graphs and coefficients

A repeated factor such as $(x-2)^2$ means the graph touches the axis at $x=2$ instead of crossing it. A cubic with positive leading coefficient runs from bottom left to top right, so it always has at least one real root.

If two polynomials are equal for all $x$, match the coefficients of each power of $x$. This is often the quickest way to find unknown constants.

Division has the form $p(x)=(x-a)q(x)+r$. Setting $x=a$ kills the quotient, which is why the remainder theorem works.

Worked examples

Common mistakes

• Evaluating $p(-a)$ instead of $p(a)$ for a divisor $x-a$.
• Forgetting that dividing by a quadratic leaves a remainder of degree up to 1.
• Stopping after finding one factor when the question asks for all roots.
• Sign slips when substituting negative values into odd powers.

Test yourself on this topic with our free practice questions.