Problem 47 - College Algebra

Question

Find each product. $$\left(4 x^{2}-1\right)^{2}$$

Step-by-Step Solution

Answer

The solution to the given exercise \((4 x^{2}-1)^{2}\) is \(16x^{4} - 8x^{2} + 1\).

Step 1: Identify the Terms

In the given binomial equation \((4x^2-1)^2\), identify \(a\) and \(b\) as in the formula. Here, \(a = 4x^2\) and \(b = 1\)

Step 2: Apply the binomial square formula

Substitute \(a\) and \(b\) into the formula for the square of a binomial. The formula \((a-b)^2 = a^2 - 2ab + b^2\), becomes \((4x^2-1)^2 = (4x^2)^2 - 2*(4x^2)*1 + (1)^2\).

Step 3: Calculate each term

Calculate each tern separately. The first term (a^2) will be \((4x^2)^2 = 16x^4\). The second term (-2ab) will be \(-2*(4x^2)*1 = -8x^2\). And the third term (b^2) will be \((1)^2 = 1\).

Step 4: Combine all terms

Combine all terms together to get the final answer. So the expression becomes \(16x^4 - 8x^2 + 1\).