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5. Algebraic Expressions
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United States of America
Grade 7

5.06 Factor with rational numbers

Lesson
Practice

Introduction

When we factor expressions with integers, we find the greatest common factor or GCF of the terms, rewrite each expression as a product of the GCF and the remaining factors, and simplify the expression by factoring out the GCF. Here's an example of this process:

\displaystyle 20x + 5\displaystyle =\displaystyle ⬚(20x + 5)The GCF of 20x and 5 is 5.
\displaystyle 20x + 5\displaystyle =\displaystyle 5(4x + 1)Factor out the GCF of 5.

Factor with decimals and fractions

Factoring expressions with decimals and fractions starts with having a common factor of the terms. We can then rewrite each term as a product of the factor and the remaining factors. Our final step will be to factor out the factor. This process is very much like factoring with integers, but now, we're working with rational numbers.

Let's look at a worked example to understand this further.

Examples

Example 1

Factor 1.6f from the following expression: 1.6 f g - 6.4f h.

Worked Solution
Create a strategy

We will factor out 1.6 from the expression, then rewrite the expression as a product of the common factor and any remaining factors in parentheses.

Apply the idea

Factoring out the given factor:

\displaystyle 1.6 f g - 6.4f h\displaystyle =\displaystyle 1.6f\times g+1.6f\times(-4h)Rewrite each term as a product of the common factor and remaining factors
\displaystyle =\displaystyle 1.6f(g+(-4h))Factor out the common factor
\displaystyle =\displaystyle 1.6f(g-4h)Adding a negative is the same as subtracting a positive
Reflect and check

We can always check to see if we have factored correctly by distributing it back out again.

\displaystyle 1.6f(g-4h)\displaystyle =\displaystyle 1.6f\times(g)+1.6f\times(-4h)
\displaystyle =\displaystyle 1.6fg-6.4fh

Example 2

Factor \dfrac{3k}{4} from the following expression: \dfrac{3}{4}jk+ \dfrac{96}{4} k

Worked Solution
Create a strategy

Factor out \dfrac{3k}{4} from the expression, then rewrite the whole expression as a product of the common factor and any remaining factors in parentheses.

Apply the idea

The fractions \dfrac{3}{4} and \dfrac{96}{4} have the same numerator and can be combined as one fraction.

\displaystyle \dfrac{3}{4}jk+ \dfrac{96}{4} k\displaystyle =\displaystyle \dfrac{3k}{4} \times j + \dfrac{3k}{4} \times 32 Rewrite each term as a product of the common factor and remaining factors
\displaystyle =\displaystyle \dfrac{3k}{4} (j + 32)Factor out the common factor
Reflect and check

We can always check to see if we have factored correctly by distributing it back out again.

\displaystyle \dfrac{3k}{4}(j+ 32)\displaystyle =\displaystyle \dfrac{3k}{4}\times(j)+\dfrac{3k}{4}\times(32)
\displaystyle =\displaystyle \dfrac{3jk}{4}+\dfrac{96k}{4}
Idea summary

When we factor expressions with rational numbers, we follow the same procedures that we would when factoring with integers:

  • Rewrite each expression as a product of the given factor and the remaining factors.
  • Simplify the expression by factoring out the factor.

Outcomes

7.EE.A.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

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