Simplify each expression. x 2 − 16 16 − x 2 \frac{x^2-16}{16-x^2}

So here we're asked to simplify the expression. Notice that we have x squared minus 16 over 16 minus x squared. Now it may be tempting to just try and break this whole thing down and factor it, but it actually turns out there's something much more simple that we can do. What I can do is change the order of either the numerator or denominator, and I'm gonna do this for the denominator. So keep the numerator the same, x squared minus 16, and then in the denominator I'm gonna put this negative x squared out front, and then we're going to have plus then we have the positive 16. Now the reason I'm writing it like this is notice we have opposite factors, we have a positive x squared and a negative 16, and then a negative x squared and a positive 16. So we know when we have opposite factors, the rule of thumb is this all just reduces down to negative one. So negative one would be the answer. As you can see, when you know how to deal with opposite factors, you don't have to go through this intensive factoring process in the numerator and denominator. You can just recognize the opposite factors and what it reduces to. Hope you found this helpful.

Simplify each expression. x 2 − 16 16 − x 2 \frac{x^2-16}{16-x^2}

16 x 2 − 16 \frac{16}{x^2-16} x 2 − 16 16

8. Rational Expressions and Functions

Simplifying Rational Expressions

Intermediate Algebra

8. Rational Expressions and Functions

Simplifying Rational Expressions

Struggling with Intermediate Algebra?

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Multiple Choice

Simplify each expression.
$\frac{x^{2} - 16}{16 - x^{2}}$

$- x^{2}$

$-1$

$x^2$

$\frac{16}{x^2-16}$

Verified step by step guidance

Identify the given expression: $\frac{x^2 - 16}{16 - x^2}$.

Recognize that both numerator and denominator are differences of squares. Recall the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$.

Factor the numerator using the difference of squares: $x^2 - 16 = (x - 4)(x + 4)$.

Factor the denominator similarly: \$16 - x^2 = (4 - x)(4 + x)$.

Notice that \$4 - x$ can be rewritten as $-(x - 4)$, so rewrite the denominator as $-(x - 4)(x + 4)$. Then, cancel the common factors $(x - 4)(x + 4)$ from numerator and denominator, leaving the simplified expression as $-1$.

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Simplify each expression.x2−1616−x2\(\frac{x^2-16}{16-x^2}\)

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Simplify each expression.
$\frac{x^{2} - 16}{16 - x^{2}}$