How to Factor $5x^{2}(x + 2)^{-\frac{1}{6}} + 10(x + 2)^{\frac{5}{6}}$ in High School Algebra

Question

Factor $5x^{2}(x + 2)^{-\frac{1}{6}} + 10(x + 2)^{\frac{5}{6}}$

Accepted Answer

Option B

Answer

Step 1: Identify the greatest common factor (GCF) of the two terms. For the algebraic part $(x+2)$, we take the term with the smaller exponent: $(x+2)^{-\frac{1}{6}}$. For the numerical coefficients, the GCF of 5 and 10 is 5. So the overall GCF is $5(x+2)^{-\frac{1}{6}}$. Step 2: Factor out the GCF from each term. For the first term $5x^{2}(x + 2)^{-\frac{1}{6}}$, dividing by $5(x+2)^{-\frac{1}{6}}$ leaves $x^2$. For the second term $10(x + 2)^{\frac{5}{6}}$, dividing by $5(x+2)^{-\frac{1}{6}}$ gives $2(x+2)^{\frac{5}{6}-(-\frac{1}{6})}=2(x+2)^{\frac{6}{6}}=2(x+2)=2x+4$. Step 3: Combine the results inside the parentheses: $x^2 + 2x + 4$. Multiply through by the factored-out GCF: $5(x+2)^{-\frac{1}{6}}(x^2 + 2x + 4)=(x+2)^{-\frac{1}{6}}(5x^2 + 10x + 20)$, which matches Option B.