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 of \((7x^3 + 5)\) and \((x^2 - 2)\) is \(7x^5 - 14x^3 + 5x^2 - 10\).

Step 1: Identify the Terms

Recognize the terms of the binomial. In the expression \((7x^3 + 5)\) and \((x^2-2)\), the terms are \(7x^3\), \(5\), \(x^2\) and \(-2\).

Step 2: Distribute the Terms

Multiply each term in the first binomial by each term in the second. This gives four multiplications: \((7x^3)(x^2)\), \((7x^3)(-2)\), \((5)(x^2)\), and \((5)(-2)\).

Step 3: Calculating the Products

Perform each multiplication. \((7x^3)(x^2) = 7x^5\), \((7x^3)(-2) = -14x^3\), \((5)(x^2) = 5x^2\), and \((5)(-2) = -10\).

Step 4: Combining Like Terms

The resulting equation from the multiplications can be written as \(7x^5 - 14x^3 + 5x^2 - 10\), however, there are no like terms, thus the expression cannot be simplified any further.