Problem 47 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The product of the given expression is \(16x^4 - 8x^2 + 1\).

Step 1: Square The first term

Take the first term of the binomial and square it to get the first term of the product. For this case, the first term is \(4x^2\), so \( (4x^2)^2\) will give us \(16x^4\) as our first term in the product.

Step 2: Square The second term

Similarly, take the second term of the binomial and square it to get the last term of the product. For this case, the second term is \(1\), so \(1^2\) will give us \(1\) as our last term in the product.

Step 3: Compute the product of 2 times the first and second terms

Next, compute the product of twice the product of the first term and the second term, which gives the middle term in the expansion of the binomial square. For this particular problem, twice the product of \(4x^2\) and \(-1\) gives us \( -8x^2 \) as our middle term.

Step 4: Sum up the Terms

Now, sum up all the terms obtained from step 1, step 2, and step 3 to find the product. Hence, the product is \(16x^4 - 8x^2 + 1\).