Problem 30 - College Algebra Essentials

Question

Find each product. $$\left(7 x^{3}+5\right)\left(x^{2}-2\right)$$

Step-by-Step Solution

Answer

\(7x^{5} - 14x^{3} + 5x^{2} - 10\)

Step 1: Distribute First Term of First Polynomial

First, distribute the first term of the first polynomial \(7x^{3}\) with each term in the second polynomial \((x^{2} - 2)\). This gives us: \(7x^{3} \cdot x^{2} - 7x^{3} \cdot 2\) which simplifies to \(7x^{5} - 14x^{3}\).

Step 2: Distribute Second Term of First Polynomial

Next, distribute the second term of the first polynomial \(5\) with each term in the second polynomial \((x^{2} - 2)\). This gives us: \(5 \cdot x^{2} - 5 \cdot 2\) which simplifies to \(5x^{2} - 10\).

Step 3: Combine Like Terms

Now, combine the results from Step 1 and Step 2 to obtain the final form of the expanded product. The final product is \(7x^{5} - 14x^{3} + 5x^{2} - 10\).