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Chapter 5: Problem 97

Factor completely. $$ 3(a+2)^{2}+30(a+2)+75 $$

Short Answer

Expert verified
3(a+7)^{2}

Step by step solution

01

Identify the Common Factor

Look at the equation 3(a+2)^{2}+30(a+2)+75 and notice that each term has a common factor of 3.
02

Factor Out the Common Factor

Factor out the common factor of 3 from each term: 3[(a+2)^{2}+10(a+2)+25]
03

Recognize the Perfect Square Trinomial

Recognize that the expression inside the bracket (a+2)^{2}+10(a+2)+25 is a perfect square trinomial. It can be written as (b^{2} + 2ab + a^{2}) where b=a+2 and a=5.
04

Factor the Perfect Square Trinomial

Factor the perfect square trinomial inside the bracket: 3[(a+2)+5]^{2}.
05

Simplify the Factorization

Simplify the factorization to get the final answer: 3(a+7)^{2}.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
When factoring trinomials, it's important to first identify any common factors among the terms. In the given exercise, each term in the expression \(3(a+2)^{2}+30(a+2)+75\) shares a common factor of 3. By factoring out this common factor, we simplify the trinomial and make it easier to handle.

After factoring out the common factor, the expression becomes \(3[(a+2)^{2}+10(a+2)+25]\). This step is critical as it reduces the complexity, allowing us to focus on the trinomial inside the brackets. Remember, identifying and factoring out common factors simplifies expressions and is an essential skill in algebra.
Perfect Square Trinomial
Next, we consider the trinomial inside the bracket: \((a+2)^{2}+10(a+2)+25\). This expression is a perfect square trinomial. A perfect square trinomial takes the form \(b^{2} + 2ab + a^{2}\), which can be factored into \((b+a)^{2}\).

In this case, we let \(b = (a+2)\) and \(a = 5\). When we substitute these values back into the trinomial, we confirm its perfect square form. Identifying a trinomial as a perfect square can greatly simplify the factoring process, making it easier to find the simplified expression.
Simplifying Expressions
Once we've factored the perfect square trinomial, we move on to simplifying the entire expression. From the previous steps, we have: \(3[(a+2)+5]^{2}\).

First, simplify inside the brackets: \(a+2+5 = a+7\). This gives us: \(3(a+7)^{2}\). The expression is now fully factored and simplified.

The final answer, \(3(a+7)^{2}\), is a much simpler form of the original expression. By methodically following factoring steps鈥攊dentifying common factors, recognizing perfect square trinomials, and performing appropriate simplifications鈥攚e can effectively break down and solve complex algebraic expressions.

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