Definition:Equating Coefficients

Definition

The technique of equating coefficients exploits the fact that identical polynomials have equal corresponding coefficients.

Examples

Quadratic

Consider the quadratic equation:

$(1): \quad x^2 + a x + b = 0$

Let the roots of $(1)$ be $\alpha$ and $\beta$.

Thus:

\(\ds x^2 + a x + b\)

\(=\)

\(\ds \paren {x - \alpha} \paren {x - \beta}\)

\(\ds \)

\(=\)

\(\ds x^2 - \paren {\alpha + \beta} x + \alpha \beta\)

Hence by equating coefficients:

\(\ds -a\)

\(=\)

\(\ds \alpha + \beta\)

as the coefficients of $x^1$ must be the same

\(\ds b\)

\(=\)

\(\ds \alpha \beta\)

as the coefficients of $x^0$ must be the same

Also see

• Results about equating coefficients can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equate coefficients
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equate coefficients