6 min readPathway sheet2-mark question; sometimes a 3-mark question with new zeroes

Finding a quadratic polynomial from its zeroes

If the zeroes have sum S and product P, the polynomial is k(x² − Sx + P), where k is any non-zero constant.

Formulak[x² − (sum of zeroes)x + (product of zeroes)], k ≠ 0

From zeroes α and βk(x − α)(x − β)

Choosing kPick k to clear fractions, e.g. k = 4 when the sum is 1/4

Why the formula works

If α and β are the zeroes, then (x − α) and (x − β) are factors. So p(x) = k(x − α)(x − β) = k[x² − (α + β)x + αβ]. Any non-zero k gives a polynomial with the same zeroes. So there are infinitely many such polynomials. That is why questions say "find a quadratic polynomial". Choose k to make the coefficients whole numbers.

Method

1
Find the sum of the zeroes, S = α + β.
2
Find the product of the zeroes, P = αβ.
3★
Write x² − Sx + P.
4
Multiply by a suitable k to clear any fractions.
5
Check: −b/a should give S and c/a should give P.

★ key step

Example 1: given the sum and product

Problem

Find a quadratic polynomial whose sum of zeroes is −3/2 and product of zeroes is −1.

Working

S = −3/2 and P = −1.
x² − Sx + P = x² − (−3/2)x + (−1) = x² + (3/2)x − 1
Multiply by k = 2 to clear the fraction: 2x² + 3x − 2.
Check: −b/a = −3/2 and c/a = −2/2 = −1.

Answer

2x² + 3x − 2

Example 2: given the zeroes

Problem

Find a quadratic polynomial whose zeroes are 3 + √2 and 3 − √2.

Working

Sum S = (3 + √2) + (3 − √2) = 6
Product P = (3 + √2)(3 − √2) = 9 − 2 = 7
x² − Sx + P = x² − 6x + 7
Check: −b/a = 6 and c/a = 7.

Answer

x² − 6x + 7

Example 3: new zeroes from old ones

Problem

If α and β are the zeroes of x² − 3x + 2, find a quadratic polynomial whose zeroes are 2α and 2β.

Working

For x² − 3x + 2: α + β = 3 and αβ = 2.
New sum = 2α + 2β = 2(α + β) = 6
New product = (2α)(2β) = 4αβ = 8
Polynomial: x² − 6x + 8
Check: the old zeroes are 1 and 2, so the new zeroes are 2 and 4. (x − 2)(x − 4) = x² − 6x + 8.

Answer

x² − 6x + 8

In real life

A rectangular rangoli border has sides adding to 7 m and an area of 12 m². Its sides are the zeroes of x² − 7x + 12, that is 3 m and 4 m.

Exam angle

2 marks: “Find a quadratic polynomial whose zeroes are −4 and 1/2.” S = −7/2, P = −2. x² + (7/2)x − 2; with k = 2: 2x² + 7x − 4. Check: (2x − 1)(x + 4).

2 marks: “Find a quadratic polynomial whose sum and product of zeroes are √3 and −2.” x² − (sum)x + product = x² − √3 x − 2.

1 marks: “Write a quadratic polynomial whose zeroes are 0 and 5.” Sum 5, product 0: x² − 5x.

Remember it

x² − (sum)x + (product): say it as "x squared, minus sum x, plus product".