Polynomial manipulation, the factor theorem and algebraic division, and the binomial expansion of (a plus b) to the power n for positive integer n using binomial coefficients.

What this dot point is asking

OCR wants you to add, subtract and multiply polynomials, divide one polynomial by a linear factor, use the factor theorem to find and confirm factors and roots, and expand $(a + b)^n$ for a positive integer $n$ using binomial coefficients $\binom{n}{r}$, including finding a single coefficient in an expansion.

The answer

Polynomial manipulation

A polynomial is a sum of terms $a_n x^n + \dots + a_1 x + a_0$. You add and subtract by collecting like terms, and multiply by expanding every pair of terms and collecting.

The factor theorem

The factor theorem links roots and factors:

To factorise a cubic, find one root by trying small values (factors of the constant term over factors of the leading coefficient), confirm with the factor theorem, divide out that factor, then factorise the resulting quadratic.

Algebraic division

Divide a polynomial by a linear factor using long division or by comparing coefficients. Comparing coefficients is often quicker: write $f(x) = (x - a)(\text{quadratic})$ with unknown coefficients and match.

The binomial theorem (positive integer power)

For a positive integer $n$:

The coefficients $\binom{n}{r}$ are the rows of Pascal's triangle. To find one specific coefficient, use the general term rather than expanding everything.

Examples in context

Reading a coefficient from an expansion

Solving an equation with a polynomial factorised

Once a cubic is fully factorised, its roots are immediate, which is why factorising is the standard route to solving cubic equations. The roots are exactly the values that make each factor zero.

Why Pascal's triangle gives the coefficients

The binomial coefficients $\binom{n}{r}$ count the number of ways of choosing $r$ factors of $b$ from the $n$ brackets when expanding $(a + b)^n$, which is why they equal the entries of Pascal's triangle. Each entry is the sum of the two above it, reflecting the identity $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$. For small powers you can read the coefficients straight off the triangle; for a single high-power coefficient, the general term is faster and avoids writing out the whole expansion.

Try this

Q1. Show that $(x - 2)$ is a factor of $x^3 - 3x^2 + 4$. [2 marks]

• Cue. $f(2) = 8 - 12 + 4 = 0$.

Q2. Find the coefficient of $x^2$ in $(1 + 2x)^6$. [2 marks]

• Cue. $\binom{6}{2}(2)^2 = 15 \times 4 = 60$.