Finding unknown coefficients with the theorems

6 minA Level3.1.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can turn a given factor or remainder into an equation, and find the unknown constants in a polynomial — including two unknowns at once.

What the question looks like

A typical question reads:

$f(x) = x^3 + ax^2 - 3x + b$ . Given that $(x - 2)$ is a factor and that $f(1) = -4$ , find $a$ and $b$ .

Each fact you are given becomes an equation:

a factor gives a value where $f$ is zero;
a remainder $c$ gives a value where $f$ equals $c$ .

Two facts give two equations, and two equations find two unknowns.

Key to the symbols

Symbol	What it means here
$a$ , $b$	the unknown coefficients you are finding
$f(k)$	the value of the polynomial at $x = k$
factor	a linear expression that divides exactly, so $f(\text{root}) = 0$
remainder	the value left over, so $f(\text{root}) = \text{remainder}$

The method

Turn each fact into $f(\text{something}) = \text{number}$ . A factor gives $0$ ; a remainder gives the stated value.
Substitute and simplify, so each fact becomes a linear equation in the unknowns.
Solve the equations together, eliminating one unknown at a time.
Check by substituting both constants back into the original facts.

The forgotten step: translating "the remainder when divided by $(x + 3)$ is 5" into $f(-3) = 5$ . The sign inside the bracket flips, and the remainder is not zero.

Worked example: Example 1: one unknown

$f(x) = x^3 + ax^2 - 3x + 2$ has remainder $8$ when divided by $x - 2$ . Find $a$ .

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Worked example: Example 2: two facts, two unknowns

$f(x) = x^3 + ax^2 + bx - 6$ . Given that $(x - 1)$ is a factor and $(x + 2)$ is a factor, find $a$ and $b$ .

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Quick check

CalculateQuestion 1 · Recall

Answer the 3 questions above to continue.