Factorization Common Factor Practice Problems & Examples

Master factorization by extracting common factors with step-by-step practice problems. Learn to convert addition expressions to multiplication form easily.

📚Master Common Factor Extraction with Interactive Practice
  • Extract common numerical factors from algebraic expressions like 2A + 4B
  • Factor out common variables from terms such as 4X + CX = X(4 + C)
  • Apply exponent laws to factor expressions like Z⁵ + 3Z⁷ = Z⁵(1 + 3Z²)
  • Handle multi-term expressions with three or more summands efficiently
  • Factor expressions in parentheses like 3A(B-5) + 8(B-5)
  • Work with opposite signs in brackets using the distributive property

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:

\( 6b^2 \)

Examples with solutions for Factorization: Common factor extraction

Step-by-step solutions included
Exercise #1

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 #2

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 #3

Break down the expression into basic terms:

3a3 3a^3

Step-by-Step Solution

To break down the expression 3a3 3a^3 , we recognize that a3 a^3 means a×a×a a \times a \times a . Therefore, 3a3 3a^3 can be decomposed as 3aaa 3 \cdot a\cdot a\cdot a .

Answer:

3aaa 3 \cdot a\cdot a\cdot a

Exercise #4

Break down the expression into basic terms:

8y 8y

Step-by-Step Solution

To break down the expression 8y 8y , we can see it as the multiplication of 8 8 and y y :

8y=8y 8y = 8 \cdot y

This shows the expression as a product of two factors, 8 8 and y y .

Answer:

8y 8\cdot y

Exercise #5

Break down the expression into basic terms:

5m 5m

Step-by-Step Solution

To break down the expression 5m 5m , we recognize it as the product of 5 5 and m m :

5m=5m 5m = 5 \cdot m

This expression can be seen as a multiplication of the constant 5 5 and the variable m m .

Answer:

5m 5\cdot m

Frequently Asked Questions

What is factorization by common factor in algebra?

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Factorization by common factor is the process of converting an expression with addition into multiplication form by extracting the largest common term. For example, 2A + 4B becomes 2(A + 2B) by extracting the common factor 2.

How do you find the greatest common factor in algebraic expressions?

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To find the greatest common factor: 1) Identify common numerical coefficients, 2) Find common variables with the lowest exponents, 3) Combine both parts. For 6A³ + 9A⁵, the GCF is 3A³.

What's the difference between factoring and the distributive property?

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Factoring extracts common factors to go from addition to multiplication (2A + 4B = 2(A + 2B)). The distributive property does the reverse, expanding multiplication into addition (2(A + 2B) = 2A + 4B).

How do you factor expressions with variables and exponents?

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Use the exponent rule aᵐⁿ = aᵐ × aⁿ to break down terms. For Z⁵ + 3Z⁷, rewrite as Z⁵ + 3Z⁵ × Z², then extract Z⁵ to get Z⁵(1 + 3Z²).

Can you factor expressions with more than two terms?

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Yes, you can factor multi-term expressions by finding the greatest common factor among all terms. For 3A³ + 6A⁵ + 9A⁴, extract 3A³ to get 3A³(1 + 2A² + 3A).

How do you handle expressions with parentheses when factoring?

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When the same expression in parentheses appears in multiple terms, treat it as the common factor. For 3A(B-5) + 8(B-5), extract (B-5) to get (B-5)(3A + 8).

What do you do when expressions have opposite signs in brackets?

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Use the fact that (X-4) = -1(4-X). For 3(X-4) + X(4-X), rewrite as 3(X-4) - X(X-4), then factor out (X-4) to get (X-4)(3-X).

How can I check if my factorization is correct?

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Use the distributive property to expand your factored form back to the original expression. If you get the same starting expression, your factorization is correct.

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