SPI Mathematics · Polynomials

Zeroes and Coefficients

The sum and the product of a polynomial's zeroes can be read straight from its coefficients, so you can build a polynomial from its zeroes or find a missing zero without solving.

All Polynomials notes

Why this matters

Nine questions in the bank: five EASY, two MODERATE and two HARD. They form a quadratic or cubic from its zeroes, find a third zero, or reason about the signs of zeroes. Each needs the sum and product rules, and a lost minus sign in −b/a is the commonest wrong option.

Concept 1 of 3: Sum and product of the zeroes of a quadratic polynomial

If α and β are the zeroes of

ax^2 + bx + c

, then it equals

a(x - \alpha)(x - \beta)

. Multiplying out shows the sum of the zeroes in the x term and their product in the constant.

Definition

$\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$ .
$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}$ and $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ .
Signs: a positive product means both zeroes have the same sign; then a positive sum means both positive, a negative sum both negative. A negative product means one positive and one negative.

Zeroes of a quadratic polynomial

\alpha + \beta = -\frac{b}{a} \qquad \alpha\beta = \frac{c}{a} \qquad \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta

Worked example

α and β are the zeroes of

2x^2 - 5x - 3

. Find

\frac{1}{\alpha} + \frac{1}{\beta}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 1 · Polynomials · Zeroes and Coefficients

The zeroes of the quadratic polynomial

x^2 + px + p

p \ne 0

are

Dropping the minus in the sum

For

2x^2 + 3x + 4

, the sum of the zeroes is

-\frac{3}{2}

and the product 2, so

\frac{1}{\alpha} + \frac{1}{\beta} = -\frac{3}{4}

, not

\frac{3}{4}

Reading signs from the product alone

A positive product says only that the zeroes share a sign. Look at the sum to tell whether both are positive or both are negative.

Drill 1 more on sum and product of the zeroes of a quadratic polynomial

Concept 2 of 3: Forming a quadratic polynomial from its zeroes

The sum and the product of the zeroes are all a quadratic needs. Put them in the pattern

x^2 - (\text{sum})x + \text{product}

, and any non-zero multiple of the result has the same zeroes.

Definition

Quadratic polynomial with zeroes α and β: $k[x^2 - (\alpha + \beta)x + \alpha\beta]$ , for any non-zero k.
So there are infinitely many polynomials with the same two zeroes.
If one zero is $p + \sqrt{q}$ and the sum is 2p, the other zero is $p - \sqrt{q}$ , and their product is $p^2 - q$ .

Quadratic from its zeroes

k\left[x^2 - (\alpha + \beta)x + \alpha\beta\right]

Worked example

Find a quadratic polynomial whose zeroes are

3 + \sqrt{2}

and

3 - \sqrt{2}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Polynomials · Zeroes and Coefficients

A quadratic polynomial, one of whose zeroes is

2 + \sqrt{5}

and sum of whose zeroes is 4, is

Thinking only one polynomial fits

x^2 - 5x + 4

2x^2 - 10x + 8

and

-x^2 + 5x - 4

all have the zeroes 1 and 4. An option that is a multiple of your answer is still right.

Adding instead of subtracting in a conjugate product

(3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7

, not

9 + 2

. The square of the root part is always subtracted.

Drill 3 more on forming a quadratic polynomial from its zeroes

Concept 3 of 3: Zeroes and coefficients of a cubic polynomial

A cubic has three zeroes, and three relations tie them to the coefficients: their sum, the sum of their products two at a time, and their product. One relation is often enough to find a missing zero.

Definition

For $ax^3 + bx^2 + cx + d$ with zeroes α, β, γ:
$\alpha + \beta + \gamma = -\frac{b}{a}$ , $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$ , $\alpha\beta\gamma = -\frac{d}{a}$ .
Two zeroes given: the sum gives the third at once.
Zeroes in an A.P.: write them $m - n$ , m, $m + n$ . Their sum is 3m, so m comes first.

Zeroes of a cubic polynomial

\alpha + \beta + \gamma = -\frac{b}{a} \qquad \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \qquad \alpha\beta\gamma = -\frac{d}{a}

Worked example

Find a cubic polynomial whose zeroes are 1, 2 and 4.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Polynomials · Zeroes and Coefficients

The third zero of the polynomial

x^3 + 7x^2 - 2x - 14

, if two of its zeroes are

\sqrt{2}

and

-\sqrt{2}

, is

Using the quadratic sign for a cubic product

For a quadratic the product of the zeroes is

+\frac{c}{a}

, but for a cubic it is

-\frac{d}{a}

. The product of the zeroes of

x^3 - 2x + 5

-5

Drill 2 more on zeroes and coefficients of a cubic polynomial

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Sum and product of the zeroes of a quadratic polynomial
Zeroes of a quadratic polynomial
$\alpha + \beta = -\frac{b}{a} \qquad \alpha\beta = \frac{c}{a} \qquad \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$
Forming a quadratic polynomial from its zeroes
Quadratic from its zeroes
$k\left[x^2 - (\alpha + \beta)x + \alpha\beta\right]$
Zeroes and coefficients of a cubic polynomial
Zeroes of a cubic polynomial
$\alpha + \beta + \gamma = -\frac{b}{a} \qquad \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \qquad \alpha\beta\gamma = -\frac{d}{a}$

Watch out for (5)

Dropping the minus in the sum
→ Sum and product of the zeroes of a quadratic polynomial
Reading signs from the product alone
→ Sum and product of the zeroes of a quadratic polynomial
Thinking only one polynomial fits
→ Forming a quadratic polynomial from its zeroes
Adding instead of subtracting in a conjugate product
→ Forming a quadratic polynomial from its zeroes
Using the quadratic sign for a cubic product
→ Zeroes and coefficients of a cubic polynomial

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