Problem 17 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The product is \(2x^3 - 9x^2 + 19x - 15\)

Step 1: Distribute the terms of the binomial to the trinomial

Multiply the first term of the binomial \((2x)\) with each term of the trinomial \((x^2, -3x, +5)\) independently and then, repeat the process with the second term of the binomial \((-3)\). This will give us: \(2x(x^2) - 2x(3x) + 2x(5) - 3(x^2) + 3(3x) - 3(5)\). From here just simplify each of the terms.

Step 2: Simplify the multiplication

After simplifying the multiplication we get: \(2x^3 - 6x^2 + 10x - 3x^2 + 9x - 15\) (Remember both \(x^2\)'s are negative and subtract to form a larger negative number).

Step 3: Combine Like Terms

Combining the like term helps to simplify the expression. Here, combine both \(x^3\) , \(x^2\), \(x\), and constant terms separately. This gives us the following expression: \(2x^3 - 9x^2 + 19x - 15\)