Problem 88 - College Algebra Essentials

Question

Perform the indicated operation or operations. $$(3 x+4)(3 x-4)\left(9 x^{2}+16\right)$$

Step-by-Step Solution

Answer

The product of the three given binomials is \( 81x^{4}-256 \).

Step 1: Use the Difference of Squares Formula

Begin by noting that the first two binomials, \( (3x+4) \) and \( (3x-4) \), form a difference of squares. According to the formula \( (a+b)(a-b)=a^{2} - b^{2} \), here \( a \) is \( 3x \) and \( b \) is \( 4 \). So, the product becomes \( (3x)^{2}-(4)^{2}=9x^{2}-16 \).

Step 2: Multiply the Outcome with the Third Binomial

Next, you multiply the result obtained from Step 1, which is \( 9x^{2}-16 \), by the third binomial \( 9x^{2}+16 \). This again forms a difference of squares. When applying the formula, here \( a \) is \( 9x^{2} \) and \( b \) is \( 16 \). So, the product becomes \( (9x^{2})^{2}-(16)^{2}=(81x^{4}-256) \).