Answer :
To factor the expression [tex]\(35x^2 + 63x^4\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, we need to identify the greatest common factor of the terms in the expression. Look at the coefficients 35 and 63:
- The GCF of 35 and 63 is 7.
Now, look at the variable part. Both terms contain a factor of [tex]\(x^2\)[/tex]:
- The first term is [tex]\(35x^2\)[/tex] and the second term is [tex]\(63x^4\)[/tex], so the common factor in terms of the variable is [tex]\(x^2\)[/tex].
Combining both, the overall GCF of the expression is [tex]\(7x^2\)[/tex].
2. Factor out the GCF:
We can factor [tex]\(7x^2\)[/tex] out of the entire expression. When we divide each term by [tex]\(7x^2\)[/tex]:
- Divide [tex]\(35x^2\)[/tex] by [tex]\(7x^2\)[/tex] to get 5.
- Divide [tex]\(63x^4\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(9x^2\)[/tex].
Therefore, factoring out the GCF from the expression, we get:
[tex]\[
35x^2 + 63x^4 = 7x^2(5 + 9x^2)
\][/tex]
3. Verify the Factorization:
To ensure that the factorization is correct, distribute [tex]\(7x^2\)[/tex] back into the parentheses:
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]
- [tex]\(7x^2 \times 9x^2 = 63x^4\)[/tex]
This matches the original expression, confirming that the factorization is correct.
So, the completely factored form of the expression [tex]\(35x^2 + 63x^4\)[/tex] is:
[tex]\[
7x^2(5 + 9x^2)
\][/tex]
1. Identify the Greatest Common Factor (GCF):
First, we need to identify the greatest common factor of the terms in the expression. Look at the coefficients 35 and 63:
- The GCF of 35 and 63 is 7.
Now, look at the variable part. Both terms contain a factor of [tex]\(x^2\)[/tex]:
- The first term is [tex]\(35x^2\)[/tex] and the second term is [tex]\(63x^4\)[/tex], so the common factor in terms of the variable is [tex]\(x^2\)[/tex].
Combining both, the overall GCF of the expression is [tex]\(7x^2\)[/tex].
2. Factor out the GCF:
We can factor [tex]\(7x^2\)[/tex] out of the entire expression. When we divide each term by [tex]\(7x^2\)[/tex]:
- Divide [tex]\(35x^2\)[/tex] by [tex]\(7x^2\)[/tex] to get 5.
- Divide [tex]\(63x^4\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(9x^2\)[/tex].
Therefore, factoring out the GCF from the expression, we get:
[tex]\[
35x^2 + 63x^4 = 7x^2(5 + 9x^2)
\][/tex]
3. Verify the Factorization:
To ensure that the factorization is correct, distribute [tex]\(7x^2\)[/tex] back into the parentheses:
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]
- [tex]\(7x^2 \times 9x^2 = 63x^4\)[/tex]
This matches the original expression, confirming that the factorization is correct.
So, the completely factored form of the expression [tex]\(35x^2 + 63x^4\)[/tex] is:
[tex]\[
7x^2(5 + 9x^2)
\][/tex]
