1Cademy - Simplifying $$\frac{y^2+y-42}{y^2-36}$$

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How to Simplify a Rational Expression

Example

Simplifying $\frac{y^2+y-42}{y^2-36}$

Simplify the rational expression $\frac{y^2+y-42}{y^2-36}$ by factoring the numerator as a trinomial of the form $y^2 + by + c$ and the denominator as a difference of squares, then canceling the common binomial factor.

Step 1 — Factor the numerator and denominator. The numerator $y^2 + y - 42$ is a trinomial requiring two numbers whose product is $-42$ and whose sum is $1$ : the pair $7$ and $-6$ works, so $y^2 + y - 42 = (y + 7)(y - 6)$ . The denominator $y^2 - 36$ is a difference of squares, since $y^2 = y^2$ and $36 = 6^2$ : $y^2 - 36 = (y + 6)(y - 6)$ . The expression becomes:

$\frac{(y + 7)(y - 6)}{(y + 6)(y - 6)}$

Step 2 — Remove the common factor. The binomial $(y - 6)$ appears in both the numerator and the denominator, so it can be divided out:

$\frac{y + 7}{y + 6}$

The simplified result is $\frac{y + 7}{y + 6}$ . Unlike the earlier example $\frac{x^2+5x+6}{x^2+8x+12}$ , where both the numerator and denominator are factored as trinomials, this example requires two different factoring techniques — trinomial factoring for the numerator and the difference of squares pattern for the denominator — before the shared binomial factor becomes visible.

Updated 2026-04-21

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OpenStax Elementary Algebra

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