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Chapter 0: Problem 16

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

Short Answer

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

Step by step solution

01

Distribute the First Term of the Binomial

Distribute the first term \(x\) of the binomial across each term in the trinomial. That is, multiply \(x\) by each term in \(\left(x^{2}-5 x+25\right)\): \(x*x^2 = x^3\), \(x*-5x=-5x^2\), \(x*25=25x\). This results in the intermediate result of \(x^3 - 5x^2 + 25x\).
02

Distribute the Second Term of the Binomial

Now, distribute the second term \(5\) of the binomial across each term in the trinomial. That is, multiply \(5\) by each term in \(\left(x^{2}-5 x+25\right)\): \(5*x^2 = 5x^2\), \(5*-5x=-25x\), \(5*25=125\). This results in the intermediate result of \(5x^2 - 25x + 125\).
03

Combine Like Terms

Now add the two results from steps 1 and 2 to get the final answer. Add the corresponding terms from the expressions obtained in the above steps: \((x^3 - 5x^2 + 25x) + (5x^2 - 25x + 125)\). Combining the like terms will give \(x^3 + 125\) as the final result.

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