Simplify \frac{x^2 - 13x + 42}{x^2 - 12x + 32} \cdot \frac{x^2 + x - 72}{54 - 3x - x^2}

Key takeaway: This is a rational-expression simplification problem. The key is to factor every polynomial completely, rewrite the last denominator with a positive leading term when helpful, and then cancel only common factors, not terms.

Step 1: Factor each polynomial

$\frac{x^2 - 13x + 42}{x^2 - 12x + 32} \cdot \frac{x^2 + x - 72}{54 - 3x - x^2}$

Factor the quadratics:

$x^2 - 13x + 42 = (x-6)(x-7)$

$x^2 - 12x + 32 = (x-4)(x-8)$

$x^2 + x - 72 = (x+9)(x-8)$

For the last denominator, factor out $-1$ first:

$54 - 3x - x^2 = -(x^2+3x-54)$

$x^2+3x-54 = (x+9)(x-6)$

So

$54 - 3x - x^2 = -(x+9)(x-6)$

Step 2: Substitute and cancel

$\frac{(x-6)(x-7)}{(x-4)(x-8)} \cdot \frac{(x+9)(x-8)}{-(x+9)(x-6)}$

Cancel common factors $(x-6)$, $(x-8)$, and $(x+9)$:

$= -\frac{x-7}{x-4}$

Final answer

$\boxed{-\frac{x-7}{x-4}}$

Restrictions: $x\neq 4,6,7,8,-9$ from the original denominators.

---

Pitfalls the pros know 👇 A common mistake is canceling across addition or subtraction terms instead of factors. For example, $x^2-12x+32$ cannot be simplified by canceling an $x$ with another $x$; it must be factored first. Also, keep track of the negative sign in $54-3x-x^2$; missing it changes the final sign.

What if the problem changes? If the last denominator had been written as $x^2+3x-54$ instead of $54-3x-x^2$, the simplified magnitude would be the same but the sign would differ because $54-3x-x^2=-(x^2+3x-54)$. For a similar expression, always check whether a factor of $-1$ is needed before canceling.

Tags: factoring quadratics, rational expressions, common factors