Question

Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [4(x2- 1)3 + 8x(x2-1)4] / [16(x2-1)3]

Accepted Answer

Start by examining the given rational expression: $$\frac{4(x^2$ - 1)^3$ + 8x(x^2$ - 1)^4$}{16(x^2$ - 1)^3$}. $$Notice that both the numerator and denominator contain powers of the expression $$(x^2 - 1$$)$$. Factor out the greatest common factor (GCF) from the numerator. The GCF is 4(x^2$ - 1)^3$, so rewrite the numerator as 4(x^2$ - 1)^3$ \left[1 + 2x(x^2$ - 1)\right]. $$Rewrite the entire expression using the factored numerator: $$\frac{4(x^2$ - 1)^3$ \left[1 + 2x(x^2$ - 1)\right]}{16(x^2$ - 1)^3$}. $$Cancel out the common factors in the numerator and denominator. Since both have $$4(x^2$$ - $$1)^3$$, divide numerator and denominator by this term to simplify the expression to $$\frac{1 + 2x(x^2$ - 1)}{4}. $$Finally, expand the term $$2x(x^2 - 1$$)$$ in the numerator to get 2x^3$ - 2x$$, then write the simplified expression as $$\frac{1 + 2x^3$ - 2x}{4}. $$This is the simplified form of the original rational expression.

Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [4(x²- 1)³ + 8x(x²-1)⁴] / [16(x²-1)³]

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Key Concepts

Factoring Polynomials

Simplifying Rational Expressions

Properties of Exponents