Algebraic Factorization of Word Problems

Factoring is the process of breaking down algebraic expressions into simpler forms by identifying common factors. It’s like finding the building blocks of an expression that, when multiplied together, gives the original expression.

In this article, we will discuss the algebraic factorization of word problems.

Basics of Factoring Algebraic Expressions

• Understand the problem and find what all the terms have in common.
• Take out this common factor and put it outside the parentheses. Divide each term by this factor and write what’s left inside the parentheses.
• Double-check your answer by multiplying the terms inside the parentheses. You should get back to the original expression.

Example

Consider the expression: (3x + 6). We can factor out the common factor, which is 3:

3x + 6 = 3(x + 2)

Here, (x + 2) is the factored form.

Solving Word Problems with Factoring

Problem 1

Jill has a rectangular garden with a length of (3x) and a width of (2x + 4). Find the algebraic expression for the area of her garden.

Solution:

The area A of the rectangular garden is given by the product of its length and width:

A=3x⋅(2x+4)

Now, let’s factor this expression:

A=3x⋅2(x+2)

So, the factored form of the area is 6x(x + 2).

Problem 2

Twice a certain number, increased by 5, is equal to three times the number decreased by 4. Find the algebraic expression for the number.

Solution:

Let the number be represented by (n). The expression for the problem is:

2n + 5 = 3n – 4

Now, let’s factor this expression:

2n + 5 = 3n – 4

2n – 3n = -4 – 5

-n = -9

n = 9

So, the algebraic expression for the number is n = 9.

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