Factoring Polynomials Worksheets - Free Printable
Educational worksheet: Factoring Polynomials Worksheets. Download and print for classroom or home learning activities.
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Polynomials Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Polynomials Worksheets
The task involves writing the prime factorization of each given monomial without using exponents. The solution is already provided in the image, but I will explain the process for each example to ensure clarity.
1. Break down the numerical coefficient into its prime factors.
2. Write out the variable part explicitly (without using exponents).
Let's go through each problem step by step:
---
- Numerical coefficient: \( 12 \)
- Prime factorization of \( 12 \): \( 12 = 2 \times 2 \times 3 \)
- Variable part: \( x^2 \)
- Write \( x^2 \) as \( x \times x \)
- Combined: \( 12x^2 = 2 \times 2 \times 3 \times x \times x \)
Answer: \( 2 \times 2 \times 3 \times x \times x \)
---
- Numerical coefficient: \( 16 \)
- Prime factorization of \( 16 \): \( 16 = 2 \times 2 \times 2 \times 2 \)
- Variable part: \( a^4 \)
- Write \( a^4 \) as \( a \times a \times a \times a \)
- Combined: \( 16a^4 = 2 \times 2 \times 2 \times 2 \times a \times a \times a \times a \)
Answer: \( 2 \times 2 \times 2 \times 2 \times a \times a \times a \times a \)
---
- Numerical coefficient: \( 9 \)
- Prime factorization of \( 9 \): \( 9 = 3 \times 3 \)
- Variable part: \( y^4 \)
- Write \( y^4 \) as \( y \times y \times y \times y \)
- Combined: \( 9y^4 = 3 \times 3 \times y \times y \times y \times y \)
Answer: \( 3 \times 3 \times y \times y \times y \times y \)
---
- Numerical coefficient: \( 15 \)
- Prime factorization of \( 15 \): \( 15 = 3 \times 5 \)
- Variable part: \( s^6 \)
- Write \( s^6 \) as \( s \times s \times s \times s \times s \times s \)
- Combined: \( 15s^6 = 3 \times 5 \times s \times s \times s \times s \times s \times s \)
Answer: \( 3 \times 5 \times s \times s \times s \times s \times s \times s \)
---
- Numerical coefficient: \( 8 \)
- Prime factorization of \( 8 \): \( 8 = 2 \times 2 \times 2 \)
- Variable part: \( n^5 \)
- Write \( n^5 \) as \( n \times n \times n \times n \times n \)
- Combined: \( 8n^5 = 2 \times 2 \times 2 \times n \times n \times n \times n \times n \)
Answer: \( 2 \times 2 \times 2 \times n \times n \times n \times n \times n \)
---
- Numerical coefficient: \( 6 \)
- Prime factorization of \( 6 \): \( 6 = 2 \times 3 \)
- Variable part: \( m^2 \)
- Write \( m^2 \) as \( m \times m \)
- Combined: \( 6m^2 = 2 \times 3 \times m \times m \)
Answer: \( 2 \times 3 \times m \times m \)
---
- Numerical coefficient: \( 10 \)
- Prime factorization of \( 10 \): \( 10 = 2 \times 5 \)
- Variable part: \( c^4 \)
- Write \( c^4 \) as \( c \times c \times c \times c \)
- Combined: \( 10c^4 = 2 \times 5 \times c \times c \times c \times c \)
Answer: \( 2 \times 5 \times c \times c \times c \times c \)
---
- Numerical coefficient: \( 18 \)
- Prime factorization of \( 18 \): \( 18 = 2 \times 3 \times 3 \)
- Variable part: \( x^2 \)
- Write \( x^2 \) as \( x \times x \)
- Combined: \( 18x^2 = 2 \times 3 \times 3 \times x \times x \)
Answer: \( 2 \times 3 \times 3 \times x \times x \)
---
- Numerical coefficient: \( 24 \)
- Prime factorization of \( 24 \): \( 24 = 2 \times 2 \times 2 \times 3 \)
- Variable part: \( z^2 \)
- Write \( z^2 \) as \( z \times z \)
- Combined: \( 24z^2 = 2 \times 2 \times 2 \times 3 \times z \times z \)
Answer: \( 2 \times 2 \times 2 \times 3 \times z \times z \)
---
- Numerical coefficient: \( 21 \)
- Prime factorization of \( 21 \): \( 21 = 3 \times 7 \)
- Variable part: \( y^3 \)
- Write \( y^3 \) as \( y \times y \times y \)
- Combined: \( 21y^3 = 3 \times 7 \times y \times y \times y \)
Answer: \( 3 \times 7 \times y \times y \times y \)
---
The solutions are already provided in the image, and they match the explanations above. Here is the boxed final answer for confirmation:
\[
\boxed{
\begin{aligned}
&12x^2: 2 \times 2 \times 3 \times x \times x \\
&16a^4: 2 \times 2 \times 2 \times 2 \times a \times a \times a \times a \\
&9y^4: 3 \times 3 \times y \times y \times y \times y \\
&15s^6: 3 \times 5 \times s \times s \times s \times s \times s \times s \\
&8n^5: 2 \times 2 \times 2 \times n \times n \times n \times n \times n \\
&6m^2: 2 \times 3 \times m \times m \\
&10c^4: 2 \times 5 \times c \times c \times c \times c \\
&18x^2: 2 \times 3 \times 3 \times x \times x \\
&24z^2: 2 \times 2 \times 2 \times 3 \times z \times z \\
&21y^3: 3 \times 7 \times y \times y \times y
\end{aligned}
}
\]
Prime Factorization Process:
1. Break down the numerical coefficient into its prime factors.
2. Write out the variable part explicitly (without using exponents).
Let's go through each problem step by step:
---
1. \( 12x^2 \)
- Numerical coefficient: \( 12 \)
- Prime factorization of \( 12 \): \( 12 = 2 \times 2 \times 3 \)
- Variable part: \( x^2 \)
- Write \( x^2 \) as \( x \times x \)
- Combined: \( 12x^2 = 2 \times 2 \times 3 \times x \times x \)
Answer: \( 2 \times 2 \times 3 \times x \times x \)
---
2. \( 16a^4 \)
- Numerical coefficient: \( 16 \)
- Prime factorization of \( 16 \): \( 16 = 2 \times 2 \times 2 \times 2 \)
- Variable part: \( a^4 \)
- Write \( a^4 \) as \( a \times a \times a \times a \)
- Combined: \( 16a^4 = 2 \times 2 \times 2 \times 2 \times a \times a \times a \times a \)
Answer: \( 2 \times 2 \times 2 \times 2 \times a \times a \times a \times a \)
---
3. \( 9y^4 \)
- Numerical coefficient: \( 9 \)
- Prime factorization of \( 9 \): \( 9 = 3 \times 3 \)
- Variable part: \( y^4 \)
- Write \( y^4 \) as \( y \times y \times y \times y \)
- Combined: \( 9y^4 = 3 \times 3 \times y \times y \times y \times y \)
Answer: \( 3 \times 3 \times y \times y \times y \times y \)
---
4. \( 15s^6 \)
- Numerical coefficient: \( 15 \)
- Prime factorization of \( 15 \): \( 15 = 3 \times 5 \)
- Variable part: \( s^6 \)
- Write \( s^6 \) as \( s \times s \times s \times s \times s \times s \)
- Combined: \( 15s^6 = 3 \times 5 \times s \times s \times s \times s \times s \times s \)
Answer: \( 3 \times 5 \times s \times s \times s \times s \times s \times s \)
---
5. \( 8n^5 \)
- Numerical coefficient: \( 8 \)
- Prime factorization of \( 8 \): \( 8 = 2 \times 2 \times 2 \)
- Variable part: \( n^5 \)
- Write \( n^5 \) as \( n \times n \times n \times n \times n \)
- Combined: \( 8n^5 = 2 \times 2 \times 2 \times n \times n \times n \times n \times n \)
Answer: \( 2 \times 2 \times 2 \times n \times n \times n \times n \times n \)
---
6. \( 6m^2 \)
- Numerical coefficient: \( 6 \)
- Prime factorization of \( 6 \): \( 6 = 2 \times 3 \)
- Variable part: \( m^2 \)
- Write \( m^2 \) as \( m \times m \)
- Combined: \( 6m^2 = 2 \times 3 \times m \times m \)
Answer: \( 2 \times 3 \times m \times m \)
---
7. \( 10c^4 \)
- Numerical coefficient: \( 10 \)
- Prime factorization of \( 10 \): \( 10 = 2 \times 5 \)
- Variable part: \( c^4 \)
- Write \( c^4 \) as \( c \times c \times c \times c \)
- Combined: \( 10c^4 = 2 \times 5 \times c \times c \times c \times c \)
Answer: \( 2 \times 5 \times c \times c \times c \times c \)
---
8. \( 18x^2 \)
- Numerical coefficient: \( 18 \)
- Prime factorization of \( 18 \): \( 18 = 2 \times 3 \times 3 \)
- Variable part: \( x^2 \)
- Write \( x^2 \) as \( x \times x \)
- Combined: \( 18x^2 = 2 \times 3 \times 3 \times x \times x \)
Answer: \( 2 \times 3 \times 3 \times x \times x \)
---
9. \( 24z^2 \)
- Numerical coefficient: \( 24 \)
- Prime factorization of \( 24 \): \( 24 = 2 \times 2 \times 2 \times 3 \)
- Variable part: \( z^2 \)
- Write \( z^2 \) as \( z \times z \)
- Combined: \( 24z^2 = 2 \times 2 \times 2 \times 3 \times z \times z \)
Answer: \( 2 \times 2 \times 2 \times 3 \times z \times z \)
---
10. \( 21y^3 \)
- Numerical coefficient: \( 21 \)
- Prime factorization of \( 21 \): \( 21 = 3 \times 7 \)
- Variable part: \( y^3 \)
- Write \( y^3 \) as \( y \times y \times y \)
- Combined: \( 21y^3 = 3 \times 7 \times y \times y \times y \)
Answer: \( 3 \times 7 \times y \times y \times y \)
---
Final Answer:
The solutions are already provided in the image, and they match the explanations above. Here is the boxed final answer for confirmation:
\[
\boxed{
\begin{aligned}
&12x^2: 2 \times 2 \times 3 \times x \times x \\
&16a^4: 2 \times 2 \times 2 \times 2 \times a \times a \times a \times a \\
&9y^4: 3 \times 3 \times y \times y \times y \times y \\
&15s^6: 3 \times 5 \times s \times s \times s \times s \times s \times s \\
&8n^5: 2 \times 2 \times 2 \times n \times n \times n \times n \times n \\
&6m^2: 2 \times 3 \times m \times m \\
&10c^4: 2 \times 5 \times c \times c \times c \times c \\
&18x^2: 2 \times 3 \times 3 \times x \times x \\
&24z^2: 2 \times 2 \times 2 \times 3 \times z \times z \\
&21y^3: 3 \times 7 \times y \times y \times y
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of factoring polynomials word problems worksheet.
