1Cademy - Simplifying $$\frac{p^3-2p^2+2p-4}{p^2-7p+10}$$

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How to Simplify a Rational Expression

Example

Simplifying $\frac{p^3-2p^2+2p-4}{p^2-7p+10}$

Simplify the rational expression $\frac{p^3-2p^2+2p-4}{p^2-7p+10}$ by factoring the numerator using the grouping method, factoring the denominator as a trinomial, and then canceling the common binomial factor.

Step 1 — Factor the numerator and denominator. The numerator $p^3 - 2p^2 + 2p - 4$ has four terms, so apply factoring by grouping. Group the first two terms and the last two terms: $(p^3 - 2p^2) + (2p - 4)$ . Factor the GCF from each group: $p^2(p - 2) + 2(p - 2)$ . Both groups share the common binomial factor $(p - 2)$ , so factor it out: $(p^2 + 2)(p - 2)$ . The denominator $p^2 - 7p + 10$ is a trinomial requiring two numbers whose product is $10$ and whose sum is $-7$ : the pair $-5$ and $-2$ works, so $p^2 - 7p + 10 = (p - 5)(p - 2)$ . The expression becomes:

$\frac{(p^2 + 2)(p - 2)}{(p - 5)(p - 2)}$

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

$\frac{p^2 + 2}{p - 5}$

The simplified result is $\frac{p^2 + 2}{p - 5}$ . Unlike the earlier examples where both the numerator and denominator were trinomials or binomials, this example requires factoring by grouping to handle the four-term numerator — a technique that splits the polynomial into two pairs, factors the GCF from each pair, and then extracts the common binomial factor. The denominator is still factored using the standard trinomial method. This combination of two different factoring techniques within a single rational expression simplification demonstrates the importance of recognizing the structure of each polynomial before choosing the appropriate factoring method.

Updated 2026-04-21

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OpenStax Elementary Algebra

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