Factorization: Common factor extraction - Examples, Exercises and Solutions

Understanding Factorization: Common factor extraction

Complete explanation with examples

Factorization

The factorization we do by extracting the common factor is our way of modifying the way the exercise is written, that is, from an expression with addition to an expression with multiplication.

For example, the expression
2A+4B2A + 4B
is composed of two terms and a plus sign. We can factor it by excluding the largest common term.
In this case it is 2 2 .

We will write it as follows:
​​​​​​​2A+4B=2×(A+2B)​​​​​​​2A + 4B = 2\times (A + 2B)

Since both terms ( A A and B B ) were multiplied by 2 2 we could "extract" it. The remaining expression is written in parentheses and the common factor (the 2 2 ) is kept out.
In this way we went from having two terms in an addition operation to having a multiplication. This procedure is called factorization.

A - Factorization

You can also apply the distributive property to do a reverse process as needed.
In certain cases we will prefer to have a multiplication and in others an addition.

Detailed explanation

Practice Factorization: Common factor extraction

Test your knowledge with 36 quizzes

Break down the expression into basic terms:

\( 5x^2 + 10 \)

Examples with solutions for Factorization: Common factor extraction

Step-by-step solutions included
Exercise #1

Break down the expression into basic terms:

2x2 2x^2

Step-by-Step Solution

The expression 2x2 2x^2 can be factored and broken down into the following basic terms:

  • The coefficient 2 2 remains as it is since it is already a basic term.
  • The term x2 x^2 can be broken down into xx x \cdot x .
  • Therefore, the entire expression can be written as 2xx 2 \cdot x \cdot x .

This breakdown helps in understanding the multiplicative nature of the expression.

Among the provided choices, the correct one that matches this breakdown is choice 2: 2xx 2\cdot x\cdot x .

Answer:

2xx 2\cdot x\cdot x

Exercise #2

Break down the expression into basic terms:

6x 6x

Step-by-Step Solution

To solve this problem, we'll clearly delineate the expression 6x 6x as follows:

  • The number 6 is the coefficient.
  • The letter x x is the variable.
  • These two components are connected by multiplication, represented as 6x 6 \cdot x .

Thus, the expression 6x 6x is equivalent to 6x 6 \cdot x , where 6 is multiplied by x x .

Examining the choice options:

  • Choice 1: 6x 6\cdot x is correct because it represents the expression as a product of the coefficient and the variable.
  • Choice 2: xxxxxx x \cdot x \cdot x \cdot x \cdot x \cdot x represents a repeated multiplication of x x , not applicable here.
  • Choice 3: 6x \frac{6}{x} represents division, not the required approach.
  • Choice 4: Incorrect, as 6x 6x can indeed be expressed as 6x 6 \cdot x .

Therefore, the best breakdown of the expression is 6x 6 \cdot x , matching choice 1.

Answer:

6x 6\cdot x

Exercise #3

Break down the expression into basic terms:

5x2 5x^2

Step-by-Step Solution

To break down the expression 5x2 5x^2 into its basic terms, we identify each component in the expression:

5is a constant multiplier 5 \, \text{is a constant multiplier}

x2 x^2 means xx x \cdot x

Therefore, 5x2 5x^2 can be rewritten as 5xx 5 \cdot x \cdot x .

Answer:

5xx 5\cdot x\cdot x

Exercise #4

Break down the expression into basic terms:

3y3 3y^3

Step-by-Step Solution

To break down the expression 3y3 3y^3 into its basic terms, we understand the components of the expression:

3is a constant multiplier 3 \, \text{is a constant multiplier}

y3 y^3 can be rewritten as yyy y \cdot y \cdot y

Thus, 3y3 3y^3 can be decomposed into 3yyy 3 \cdot y \cdot y \cdot y .

Answer:

3yyy 3\cdot y\cdot y \cdot y

Exercise #5

Break down the expression into basic terms:

4a2 4a^2

Step-by-Step Solution

To break down the expression 4a2 4a^2 into basic terms, we need to look at each factor:

4is a constant multiplier 4 \, \text{is a constant multiplier}

a2 a^2 means aa a \cdot a

Hence, 4a2 4a^2 is equivalent to 4aa 4 \cdot a \cdot a .

Answer:

4aa 4\cdot a\cdot a

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