Technique

Equate coefficients of like powers

Algebra • Polynomial

Statement

Two polynomials in

x

take the same value for every

x

if and only if the coefficients of each power of

x

are equal. So an unknown polynomial of assumed shape can be found by expanding the assumed form, equating the coefficient of each power with the corresponding coefficient of the target - including every power whose coefficient on the target is

0

- and solving the resulting equations.

How to use it

1Write the assumed form with the right degree first: squaring a polynomial doubles its degree, so the square root of a quartic is sought as a quadratic.
2Start from the ends - the highest power and the constant term - because those equations usually involve one unknown each.
3A power missing from the target has coefficient $0$ , not 'no equation'; that zero is as much a condition as any other.
4Count the equations against the unknowns: any coefficient left over after the unknowns are pinned down is a free consistency check, and it must come out right.

Worked example

Find

a

and

b

such that

(a x + b)^{2} = 9 x^{2} - 12 x + 4

for every

x

Step-by-step

1
$(a x + b)^{2} = a^{2} x^{2} + 2 ab x + b^{2}$
Why: Expand the assumed form and sort by power of $x$ .
2
$a^{2} = 9 \Rightarrow a = 3$ ; then $2 ab = - 12 \Rightarrow 6 b = - 12 \Rightarrow b = - 2$
Why: Take the positive value of $a$ ; the negative choice flips the sign of the whole bracket.
3
Check on the constant: $b^{2} = (- 2)^{2} = 4$
Why: That coefficient was not used to find $a$ and $b$ , so it is a genuine check.

Answer

a = 3, b = - 2

Watch out

Equating coefficients needs the identity to hold for every

x

; for an equation true at only some values of

x

it is invalid, and matching coefficients there will produce answers the equation does not have. A square-root problem has two sign choices for the leading coefficient, and the convention is to take the positive one; the other unknowns are then forced, not free.

Builds on

Square of a sum of several terms

Also known as

compare coefficients, undetermined coefficients, equate coefficients of like powers, polynomial identity, method of undetermined coefficients

Appears in past papers

2020 (II)

Q35