Question

question #19 which of the following is a common factor of the rational expression $\frac{2x^{2}+13x + 20}{4x^{2}-64}$ i. $x - 4$ ii. $x + 4$ iii. $2x + 5$ iv. there are no common factors iv only i, ii, and iii ii only i only

Explanation:

Step1: Factor the numerator

Factor $2x^{2}+13x + 20$. We need two numbers that multiply to $2\times20 = 40$ and add up to $13$. The numbers are $8$ and $5$. So, $2x^{2}+13x + 20=2x^{2}+8x+5x + 20=2x(x + 4)+5(x + 4)=(2x + 5)(x + 4)$.

Step2: Factor the denominator

Factor $4x^{2}-64$. First, factor out the common - factor of $4$: $4x^{2}-64 = 4(x^{2}-16)$. Then, use the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$ with $a=x$ and $b = 4$. So, $4(x^{2}-16)=4(x + 4)(x - 4)$.

Step3: Identify common factors

The rational expression $\frac{2x^{2}+13x + 20}{4x^{2}-64}=\frac{(2x + 5)(x + 4)}{4(x + 4)(x - 4)}$. The common factor is $x + 4$.

Answer:

II. $x + 4$