Problem 16 - College Algebra

Question

Find each product. $$(x+5)\left(x^{2}-5 x+25\right)$$

Step-by-Step Solution

Answer

The product of \((x+5)\) and \(\left(x^{2}-5 x+25\right)\) is \(x^3 + 125\).

Step 1: Distribute the terms of the polynomial (x+5) across each term in the polynomial \((x^{2}-5x+25)\)

Do the following operations: \(x \cdot x^2\), \(x \cdot -5x\), \(x \cdot 25\), \(5 \cdot x^2\), \(5 \cdot -5x\), and \(5 \cdot 25 \). This will give the six products: \(x^3\), \(-5x^2\), \(25x\), \(5x^2\), \(-25x\), and \(125\). Write these down.

Step 2: Combine like terms

We have terms with \(x^3\), \(x^2\), \(x\), and a constant. Combine each of these. The term with \(x^3\): \(x^3\), there's no other \(x^3\) term so remains the same. The terms with \(x^2\): \(-5x^2 + 5x^2\) cancels out to zero. The terms with \(x\): \(25x - 25x\), this also cancels out to zero. Finally, the constant term is \(125\). So in standard form, the product is \(x^3 + 125\).

Step 3: Write the final answer

The final product is \(x^3 + 125\). This is the simplest form without like terms and is in standard form.