Problem 30 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Distribute Each Term

First, distribute each term of the first polynomial \((7 x^{3}+5)\) to each term of the second polynomial \((x^{2}-2)\). This will give four products because there are two terms in each polynomial.

Step 2: Calculate First Set of Products

Multiply the \(7x^3\) by each term in the second polynomial. So, \(7x^{3}*x^{2}\) gives \(7x^{5}\) and \(7x^{3}*-2\) gives \(-14x^{3}\). So at this stage, the equation looks like this: \(7x^{5}-14x^{3}+...\)

Step 3: Calculate Second Set of Products

Next, multiply the 5 (second term in the first polynomial) by each term in the second polynomial. So, \(5*x^{2}\) gives \(5x^{2}\) and \(5*-2\) gives \(-10\). After this, equation looks like this: \(7x^{5}-14x^{3}+5x^{2}-10\)

Step 4: Write Out The Full Equation

Now write out the full equation with all four products: \(7x^{5}-14x^{3}+5x^{2}-10\)