Polynomials

Polynomials is usually worth about 4–7 marks: a multiple choice question on a graph or the roots, plus a Section II part using the remainder theorem or sums and products of zeros. It builds on the quadratics and graph sketching in the Advanced focus area Working with Functions.

• Language and graphs: degree, leading term and coefficient, monic, constant term, the zero polynomial, end behaviour, zeros and multiplicity, sketching from factored form.
• Division: write $P(x)=A(x)Q(x)+R(x)$, where the remainder has lower degree than the divisor.
• Remainder and factor theorems: prove and use them; factorise cubics and quartics.
• Sums and products of zeros: prove the results for quadratics, cubics and quartics and use them to evaluate symmetric expressions.

A polynomial of degree $n$ is

where $n$ is a whole number and $a_n\ne 0$. The leading term is $a_n{x}^n$, the leading coefficient is $a_n$, and $a_0$ is the constant term. A polynomial is monic if $a_n=1$. The zero polynomial $P(x)=0$ has no degree and no leading term.

When you add two polynomials, the degree is at most the larger degree, and can be smaller if the leading terms cancel. For example $(2x^3+x)+(-2x^3+5)$ has degree 1.

End behaviour

For large $|x|$ the leading term dominates. Even degree with $a_n>0$: both ends go up. Odd degree with $a_n>0$: down on the left, up on the right. A negative leading coefficient flips these.

Zeros and multiplicity

A zero of $P(x)$ is a value $\alpha$ with $P(\alpha )=0$; a root is a solution of the equation $P(x)=0$. If $(x-\alpha {)}^k$ is a factor but $(x-\alpha {)}^{k+1}$ is not, $\alpha$ is a zero of multiplicity $k$. The multiplicity tells you the shape at the $x$-intercept:

In general, odd multiplicity crosses the axis, even multiplicity bounces off it.

To sketch a factored polynomial: find the $y$-intercept, mark the zeros with their multiplicities, decide the end behaviour from the leading term, then join up. For example $y=-x(x-2{)}^2$ has degree 3 with negative leading coefficient, so it starts top left, crosses at $x=0$ and touches at $x=2$, finishing bottom right.

Long division of polynomials works like long division of numbers. The result is written

where $A(x)$ is the divisor, $Q(x)$ the quotient and $R(x)$ the remainder, with $\mathrm{deg}R<\mathrm{deg}A$ (or $R=0$). Dividing by a linear divisor $(x-\alpha )$ therefore leaves a constant remainder $r$.

Remainder theorem. The remainder when $P(x)$ is divided by $(x-\alpha )$ is $P(\alpha )$.

Proof: $P(x)=(x-\alpha )Q(x)+r$. Substitute $x=\alpha$: $P(\alpha )=0\cdot Q(\alpha )+r=r$.

Factor theorem. $(x-\alpha )$ is a factor of $P(x)$ if and only if $P(\alpha )=0$. This follows because the remainder is zero exactly when $P(\alpha )=0$.

To factorise a cubic with integer coefficients, test integer values of $\alpha$ that divide the constant term, then divide to find the quadratic factor.

Worked example 1: factorising a cubic

Factorise $P(x)=x^3-4x^2+x+6$ and solve $P(x)=0$.

The constant term is 6, so try $\pm 1,\pm 2,\pm 3,\pm 6$:

So $(x+1)$ is a factor. Dividing $P(x)$ by $(x+1)$ gives quotient $x^2-5x+6$ with remainder 0, so

The roots are $x=-1,2,3$.

Worked example 2: finding unknown coefficients

$P(x)=x^3+a{x}^2+bx-6$. $(x-2)$ is a factor, and the remainder on division by $(x+1)$ is $-12$. Find $a$ and $b$.

Use the factor theorem and the remainder theorem to get two equations:

These simplify to $2a+b=-1$ and $a-b=-5$. Adding them gives $3a=-6$, so $a=-2$ and then $b=3$.

Extension of the idea. When dividing by a quadratic with two distinct linear factors, write the remainder as $R(x)=mx+c$ and substitute both zeros. For example, if $P(1)=4$ and $P(3)=10$, then dividing $P(x)$ by $(x-1)(x-3)$ gives $m+c=4$ and $3m+c=10$, so the remainder is $3x+1$.

If $\alpha$ and $\beta$ are the zeros of $a{x}^2+bx+c$, then $a{x}^2+bx+c=a(x-\alpha )(x-\beta )$. Expanding and comparing coefficients proves the results. The same method gives the cubic and quartic results.

For $a{x}^3+b{x}^2+cx+d$ with zeros $\alpha ,\beta ,\gamma$:

For $a{x}^4+b{x}^3+c{x}^2+dx+e$ with zeros $\alpha ,\beta ,\gamma ,\delta$:

Here $\sum \alpha \beta$ means the sum of the products of the zeros taken two at a time (six terms for a quartic). The signs alternate, starting with minus. The current reference sheet lists only the cubic results, and the sheet for the new syllabus may differ, so learn all of them: the syllabus requires you to prove the quadratic, cubic and quartic cases.

Useful identities for symmetric expressions:

Worked example 3: symmetric functions of the zeros

The zeros of $2x^3-3x^2+4x+5$ are $\alpha$, $\beta$ and $\gamma$. Find ${\alpha }^2+{\beta }^2+{\gamma }^2$ and $\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }$, and explain why the polynomial cannot have three real zeros.

Here $a=2$, $b=-3$, $c=4$, $d=5$:

Then

If all three zeros were real, the sum of their squares would be at least 0. It is negative, so at least one zero is not real. (A cubic always has at least one real zero, so here there is exactly one.)

• Sign errors in the sums and products: the sum of the zeros is $-\frac{b}{a}$, not $\frac{b}{a}$; for a cubic the product is $-\frac{d}{a}$, for a quartic it is $+\frac{e}{a}$.
• Forgetting to divide by the leading coefficient $a$ when the polynomial is not monic.
• Substituting $x=2$ instead of $x=-2$ when the divisor is $(x+2)$.
• Saying the degree of a sum is the larger degree without checking whether the leading terms cancel.
• Drawing a double zero crossing the axis, or a single zero bouncing off it.
• Writing the remainder on division by a quadratic as a constant. It can be linear, $mx+c$.

• Show that $(x-\alpha )$ is a factor: evaluate $P(\alpha )$, show it equals 0, and quote the factor theorem by name.
• Lay out long division in full or state the quotient clearly. Markers want to see how you found the quadratic factor.
• For questions on symmetric expressions, write down $\sum \alpha$, $\sum \alpha \beta$ and $\alpha \beta \gamma$ first, each with its sign. Most of the marks come from getting these right.
• For a sketch, show all intercepts, the behaviour at each zero (cross, touch or flatten) and the end behaviour. You do not need exact turning points unless asked.
• If a question gives a relationship between the zeros (for example one zero is the negative of another), write the zeros as $\alpha$, $-\alpha$, $\beta$ and use the sum and product equations to solve for them.
• Proofs of the sum and product results need the expansion of $a(x-\alpha )(x-\beta )(x-\gamma )$ and a statement that you are equating coefficients.

• Advanced, Working with Functions: quadratics, the discriminant and the sum and product of the roots of a quadratic are assumed knowledge.
• Further work with functions (ME11-FN): cubic inequalities use the factored polynomial and its graph; reciprocal graphs of polynomials put asymptotes at the zeros.
• The binomial theorem (ME11-BT): expanding powers like $(x+a{)}^n$ produces polynomials whose coefficients are binomial coefficients.
• Advanced, Differential Calculus: finding the stationary points of a polynomial confirms the turning points and inflections that multiplicity predicts.