Problem 49 - College Algebra

Question

Find each product. $$(7-2 x)^{2}$$

Step-by-Step Solution

Answer

The multiplication of \( (7-2x)^2 \) results in \( 4x^2 -28x + 49 \)

Step 1: Understand the Squaring of Binomial

When you see \( (7-2x)^2 \), remember that it means \( (7-2x) * (7-2x) \). Do not make the common mistake of simply squaring each term individually, instead you should do the actual multiplication operation.

Step 2: Apply FOIL Method

FOIL stands for First, Outer, Inner, and Last, which is a method for multiplying binomials. For our specific problem, the FOIL method would look like this: First: 7*7 = \( 49 \). Outer: 7*(-2x) = \( -14x \). Inner: -2x*7 = \( -14x \). Last: -2x*-2x = \( 4x^2 \). So the intermediate result by adding all these together gives \( 49 - 14x -14x + 4x^2 \).

Step 3: Simplify the Result

Combine like terms from the intermediate result. This gives \( 49 -28x + 4x^2 \). Since the degree of \( x^2 \) is the highest, by convention, we write it at the front, so our polynomial becomes: \( 4x^2 -28x + 49\).