Answer :
Sure, let's break down how to factor the expression [tex]\( 21x^4 + 70x \)[/tex] completely, step by step.
1. Identify the Greatest Common Factor (GCF):
The first step in factoring the expression is to identify the greatest common factor of the terms.
- The coefficients are 21 and 70. The GCF of 21 and 70 is 7.
- The variable terms are [tex]\( x^4 \)[/tex] and [tex]\( x \)[/tex]. The GCF of [tex]\( x^4 \)[/tex] and [tex]\( x \)[/tex] is [tex]\( x \)[/tex] (since [tex]\( x \)[/tex] is the lowest power).
2. Factor out the GCF:
Now, we factor out 7[tex]\( x \)[/tex] from each term in the expression.
- [tex]\( 21x^4 \)[/tex] divided by 7[tex]\( x \)[/tex] is [tex]\( 3x^3 \)[/tex]
- [tex]\( 70x \)[/tex] divided by 7[tex]\( x \)[/tex] is 10
So the expression becomes:
[tex]\[
7x (3x^3 + 10)
\][/tex]
3. Check for further factorization:
We need to see if the binomial [tex]\( 3x^3 + 10 \)[/tex] can be factored further.
- In this case, [tex]\( 3x^3 + 10 \)[/tex] does not factor further since it does not have any common factors and is not a special form like a sum or difference of cubes.
Therefore, the completely factored form of the expression [tex]\( 21x^4 + 70x \)[/tex] is:
[tex]\[
7x(3x^3 + 10)
\][/tex]
And that is the final answer!
1. Identify the Greatest Common Factor (GCF):
The first step in factoring the expression is to identify the greatest common factor of the terms.
- The coefficients are 21 and 70. The GCF of 21 and 70 is 7.
- The variable terms are [tex]\( x^4 \)[/tex] and [tex]\( x \)[/tex]. The GCF of [tex]\( x^4 \)[/tex] and [tex]\( x \)[/tex] is [tex]\( x \)[/tex] (since [tex]\( x \)[/tex] is the lowest power).
2. Factor out the GCF:
Now, we factor out 7[tex]\( x \)[/tex] from each term in the expression.
- [tex]\( 21x^4 \)[/tex] divided by 7[tex]\( x \)[/tex] is [tex]\( 3x^3 \)[/tex]
- [tex]\( 70x \)[/tex] divided by 7[tex]\( x \)[/tex] is 10
So the expression becomes:
[tex]\[
7x (3x^3 + 10)
\][/tex]
3. Check for further factorization:
We need to see if the binomial [tex]\( 3x^3 + 10 \)[/tex] can be factored further.
- In this case, [tex]\( 3x^3 + 10 \)[/tex] does not factor further since it does not have any common factors and is not a special form like a sum or difference of cubes.
Therefore, the completely factored form of the expression [tex]\( 21x^4 + 70x \)[/tex] is:
[tex]\[
7x(3x^3 + 10)
\][/tex]
And that is the final answer!
