Polynomials — chapter notes

A polynomial in $x$ is a sum of terms $ax^n$ where $n$ is a whole number ($0, 1, 2, \dots$). The highest power with a non-zero coefficient is the degree.

• Degree $1$: linear, $ax + b$ ($a \ne 0$).
• Degree $2$: quadratic, $ax^2 + bx + c$ ($a \ne 0$).
• Degree $3$: cubic, $ax^3 + bx^2 + cx + d$ ($a \ne 0$).

$x^2 + \dfrac1x$ and $\sqrt x + 1$ are not polynomials (powers $-1$ and $\dfrac12$).

The value of $p(x)$ at $x = k$ is $p(k)$, found by substitution. A real number $k$ is a zero of $p(x)$ if $p(k) = 0$.

The zero of a linear polynomial $ax + b$ is $-\dfrac ba$.

To test a zero, substitute: $p(x) = x^2 - 3x + 2$ gives $p(2) = 4 - 6 + 2 = 0$, so $2$ is a zero. A known zero gives an equation for an unknown coefficient: if $3$ is a zero of $x^2 + kx + 15$, then $9 + 3k + 15 = 0$ and $k = -8$.

The zeroes of $p(x)$ are the $x$-coordinates of the points where the graph of $y = p(x)$ meets the $x$-axis.

• Linear: a straight line, exactly one zero.
• Quadratic: a parabola, opening upward if $a > 0$ and downward if $a < 0$. It can cross the $x$-axis twice (two zeroes), touch it once (one zero), or not meet it (no zero).
• The $y$-intercept is $p(0)$, the constant term — not a zero.

A polynomial of degree $n$ has at most $n$ zeroes, because its graph meets the $x$-axis at most $n$ times.

• Cubic: $1$, $2$ or $3$ zeroes (at least one, because its two ends go in opposite directions).
• The graph of $x^4 - 5x^2 + 4$ meets the axis at $\pm1$ and $\pm2$: four zeroes.

To count zeroes from a graph, count the meeting points with the $x$-axis — not the turning points, and not the $y$-intercept.

In context, keep only meaningful zeroes: for $h = -5t^2 + 10t + 15$ the zeroes are $-1$ and $3$, and the ball lands at $t = 3$ s.

To factorise $ax^2 + bx + c$: find two numbers with product $ac$ and sum $b$, split $bx$ into two terms with them, and factorise in pairs.

Example: $10x^2 - 3x - 1$: $ac = -10 = (-5)(2)$, $-5 + 2 = -3$. So $10x^2 - 5x + 2x - 1 = (5x + 1)(2x - 1)$, zeroes $-\dfrac15$ and $\dfrac12$.

Each factor $(px + q)$ gives the zero $-\dfrac qp$. Difference of squares: $9t^2 - 16 = (3t - 4)(3t + 4)$.

If $\alpha$ and $\beta$ are the zeroes of $ax^2 + bx + c$ ($a \ne 0$), then

Why: $ax^2 + bx + c = a(x - \alpha)(x - \beta) = ax^2 - a(\alpha + \beta)x + a\alpha\beta$; compare coefficients.

Use it to verify factorisations, to read the sum and product without factorising, and to find unknown coefficients from conditions (reciprocal zeroes ⇒ product $1$; opposite zeroes ⇒ sum $0$).

Not in the 2026–27 syllabus: the relationship for cubic polynomials, and the division algorithm.

If the sum of the zeroes is $S$ and the product is $P$, then every quadratic with these zeroes is

Choose $k$ to clear fractions. Examples: zeroes $-3, 4$ → $x^2 - x - 12$; sum $-\dfrac32$, product $-1$ → $2x^2 + 3x - 2$; zeroes $3 \pm \sqrt2$ → $x^2 - 6x + 7$.

New zeroes from old: if $\alpha, \beta$ are zeroes of $x^2 - 7x + 10$, the polynomial with zeroes $\alpha^2, \beta^2$ has sum $\alpha^2 + \beta^2 = 29$ and product $(\alpha\beta)^2 = 100$: $x^2 - 29x + 100$.

Expressions unchanged by swapping $\alpha$ and $\beta$ can be evaluated from the sum and product, without finding the zeroes:

• $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$
• $\dfrac1\alpha + \dfrac1\beta = \dfrac{\alpha + \beta}{\alpha\beta}$
• $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$
• $\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta)$
• $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$

Application: sides of a rectangle as zeroes of $x^2 - 17x + 60$: perimeter $34$ m, area $60$ m², diagonal $\sqrt{17^2 - 120} = 13$ m.