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Answer :
To solve the given problem of factoring the expression [tex]\( 35x^5 - 28x^3 - 21x \)[/tex], we need to first find the greatest common factor (GCF) of all the terms.
Step 1: Identify the GCF
1. Coefficients: Look at the numbers in front of [tex]\( x \)[/tex] in each term: 35, 28, and 21.
- The GCF of 35, 28, and 21 is 7. We find this by determining the largest number that divides all three coefficients evenly.
2. Variables: Look at the powers of [tex]\( x \)[/tex] in each term: [tex]\( x^5 \)[/tex], [tex]\( x^3 \)[/tex], and [tex]\( x \)[/tex].
- The GCF here is [tex]\( x \)[/tex] because [tex]\( x \)[/tex] is the lowest power of [tex]\( x \)[/tex] common across all terms.
Combining these, the GCF of the terms in the expression is [tex]\( 7x \)[/tex].
Step 2: Factor out the GCF
Divide each term in the expression [tex]\( 35x^5 - 28x^3 - 21x \)[/tex] by the GCF [tex]\( 7x \)[/tex]:
- For [tex]\( 35x^5 \)[/tex], dividing by [tex]\( 7x \)[/tex] gives [tex]\( 5x^4 \)[/tex].
- For [tex]\( 28x^3 \)[/tex], dividing by [tex]\( 7x \)[/tex] gives [tex]\( 4x^2 \)[/tex].
- For [tex]\( 21x \)[/tex], dividing by [tex]\( 7x \)[/tex] gives 3.
So, the expression becomes:
[tex]\[ 35x^5 - 28x^3 - 21x = 7x(5x^4 - 4x^2 - 3) \][/tex]
Step 3: Verification by Multiplication
To check if our factoring is correct, expand [tex]\( 7x(5x^4 - 4x^2 - 3) \)[/tex] back out:
- Distribute [tex]\( 7x \)[/tex] across each term inside the parentheses:
- [tex]\( 7x \cdot 5x^4 = 35x^5 \)[/tex]
- [tex]\( 7x \cdot (-4x^2) = -28x^3 \)[/tex]
- [tex]\( 7x \cdot (-3) = -21x \)[/tex]
After performing these multiplications, you get back the original expression:
[tex]\[ 35x^5 - 28x^3 - 21x \][/tex]
This confirms the factoring is correct. Therefore, the expression [tex]\( 35x^5 - 28x^3 - 21x \)[/tex] factored out by the greatest common factor is:
[tex]\[ 7x(5x^4 - 4x^2 - 3) \][/tex]
Step 1: Identify the GCF
1. Coefficients: Look at the numbers in front of [tex]\( x \)[/tex] in each term: 35, 28, and 21.
- The GCF of 35, 28, and 21 is 7. We find this by determining the largest number that divides all three coefficients evenly.
2. Variables: Look at the powers of [tex]\( x \)[/tex] in each term: [tex]\( x^5 \)[/tex], [tex]\( x^3 \)[/tex], and [tex]\( x \)[/tex].
- The GCF here is [tex]\( x \)[/tex] because [tex]\( x \)[/tex] is the lowest power of [tex]\( x \)[/tex] common across all terms.
Combining these, the GCF of the terms in the expression is [tex]\( 7x \)[/tex].
Step 2: Factor out the GCF
Divide each term in the expression [tex]\( 35x^5 - 28x^3 - 21x \)[/tex] by the GCF [tex]\( 7x \)[/tex]:
- For [tex]\( 35x^5 \)[/tex], dividing by [tex]\( 7x \)[/tex] gives [tex]\( 5x^4 \)[/tex].
- For [tex]\( 28x^3 \)[/tex], dividing by [tex]\( 7x \)[/tex] gives [tex]\( 4x^2 \)[/tex].
- For [tex]\( 21x \)[/tex], dividing by [tex]\( 7x \)[/tex] gives 3.
So, the expression becomes:
[tex]\[ 35x^5 - 28x^3 - 21x = 7x(5x^4 - 4x^2 - 3) \][/tex]
Step 3: Verification by Multiplication
To check if our factoring is correct, expand [tex]\( 7x(5x^4 - 4x^2 - 3) \)[/tex] back out:
- Distribute [tex]\( 7x \)[/tex] across each term inside the parentheses:
- [tex]\( 7x \cdot 5x^4 = 35x^5 \)[/tex]
- [tex]\( 7x \cdot (-4x^2) = -28x^3 \)[/tex]
- [tex]\( 7x \cdot (-3) = -21x \)[/tex]
After performing these multiplications, you get back the original expression:
[tex]\[ 35x^5 - 28x^3 - 21x \][/tex]
This confirms the factoring is correct. Therefore, the expression [tex]\( 35x^5 - 28x^3 - 21x \)[/tex] factored out by the greatest common factor is:
[tex]\[ 7x(5x^4 - 4x^2 - 3) \][/tex]
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