Problem 47 - Algebra 1: Concepts and Skills

Question

COMMON FACTOR Factor the expression. $$ 4 b^{2}-40 b+100 $$

Step-by-Step Solution

Answer

The factored form of the expression \(4b^{2}-40b+100\) is \(4(b-5)^{2}\).

Step 1: Identify Common Factor

In order to factor the expression \(4b^{2}-40b+100\), first look for a common factor present in all terms. In this case, that common factor is 4. Divide each term by 4.

Step 2: Factor Out Common Factor

Factor out the common factor 4 from the expression, giving \(4(b^{2}-10b+25)\).

Step 3: Recognize Perfect Square Trinomial

Within the parentheses now is a perfect square trinomial, which is in the form of \(a^{2}-2ab+b^{2}\). This can be factored into \((a-b)^{2}\). In this case, \(a\) is equal to \(b\) and \(b\) is equal to 5.

Step 4: Factor the Perfect Square Trinomial

Factor the expression within the parentheses. In this case, \(b^{2}-10b+25 = (b-5)^{2}\).

Key Concepts

Common Factor

When you look at an algebraic expression like \(4b^2 - 40b + 100\), one of the first things you should do is check if there's a common factor among the terms. A common factor is a number or variable that can evenly divide each term in the expression. In our example, the common factor is 4, because:

• \(4b^2\) is divisible by 4
• \(40b\) is divisible by 4
• 100 is divisible by 4

Finding the common factor helps simplify the expression, making the subsequent steps easier. To factor out the common factor, divide each term by 4, resulting in:
\[4(b^2 - 10b + 25)\]
This step not only simplifies the expression but also uncovers what's next, leading you to the core of the problem.

Perfect Square Trinomial

The expression within the parentheses, \(b^2 - 10b + 25\), is known as a perfect square trinomial. A perfect square trinomial is a special form of quadratic expression that can be rewritten as the square of a binomial.
The general form is:

• \(a^2 - 2ab + b^2 = (a-b)^2\)
• \(a^2 + 2ab + b^2 = (a+b)^2\)

In our case:
\[b^2 - 10b + 25 = (b-5)^2\]
Here, \(a = b\) and \(b = 5\), both found by comparing the expression to the general formula. Recognizing this form allows for quick and straightforward factoring, which is especially useful for solving or simplifying algebraic equations.

Algebraic Factorization

Algebraic factorization is the process of breaking down an expression into a product of simpler expressions or factors. This is a crucial step in algebra as it simplifies expressions and aids in solving equations. The process often involves several techniques, one of which is identifying and factoring out the common factor, as seen in our example.
Factorizing not only involves using common factors but also recognizing patterns such as perfect square trinomials. In the example \(4b^2 - 40b + 100\), we first factored out 4 and then dealt with a perfect square trinomial. Thus, factorization resulted in:
\[4(b-5)^2\]
In algebra, these steps are critical as they help in simplifying complex expressions and solving equations efficiently. Getting a good grip on these techniques helps develop a strong foundation in algebra.