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Answer :
Sure! Let's go through the steps to factor the expression [tex]\(30x - 70x^4\)[/tex] completely.
1. Identify the Greatest Common Factor (GCF):
Look at the coefficients and the powers of [tex]\(x\)[/tex] in each term of the expression:
- The terms are [tex]\(30x\)[/tex] and [tex]\(-70x^4\)[/tex].
- The GCF of the coefficients 30 and 70 is 10.
- The smallest power of [tex]\(x\)[/tex] is [tex]\(x^1\)[/tex]. Thus, the GCF of the terms that includes the variable is [tex]\(x\)[/tex].
Therefore, the GCF of [tex]\(30x\)[/tex] and [tex]\(-70x^4\)[/tex] is [tex]\(10x\)[/tex].
2. Factor out the GCF:
Divide each term in the expression by the GCF [tex]\(10x\)[/tex] and factor it out:
[tex]\[
30x \div 10x = 3
\][/tex]
[tex]\[
-70x^4 \div 10x = -7x^3
\][/tex]
So, when you factor out [tex]\(10x\)[/tex], the expression inside the parentheses will be:
[tex]\[
10x(3 - 7x^3)
\][/tex]
3. Check for further factorization:
Look at the expression within the parentheses, [tex]\(3 - 7x^3\)[/tex]. This expression cannot be factored further using real numbers since it does not have any common factors or obvious patterns (like a difference of squares or cubes) that can be factored.
Thus, the complete factorization of the expression [tex]\(30x - 70x^4\)[/tex] is:
[tex]\[
-10x(7x^3 - 3)
\][/tex]
Note: The factors can also be expressed as [tex]\(10x(-7x^3 + 3)\)[/tex] which is equivalent to [tex]\(-10x(7x^3 - 3)\)[/tex]. Both forms are valid and correctly factor the original expression completely.
1. Identify the Greatest Common Factor (GCF):
Look at the coefficients and the powers of [tex]\(x\)[/tex] in each term of the expression:
- The terms are [tex]\(30x\)[/tex] and [tex]\(-70x^4\)[/tex].
- The GCF of the coefficients 30 and 70 is 10.
- The smallest power of [tex]\(x\)[/tex] is [tex]\(x^1\)[/tex]. Thus, the GCF of the terms that includes the variable is [tex]\(x\)[/tex].
Therefore, the GCF of [tex]\(30x\)[/tex] and [tex]\(-70x^4\)[/tex] is [tex]\(10x\)[/tex].
2. Factor out the GCF:
Divide each term in the expression by the GCF [tex]\(10x\)[/tex] and factor it out:
[tex]\[
30x \div 10x = 3
\][/tex]
[tex]\[
-70x^4 \div 10x = -7x^3
\][/tex]
So, when you factor out [tex]\(10x\)[/tex], the expression inside the parentheses will be:
[tex]\[
10x(3 - 7x^3)
\][/tex]
3. Check for further factorization:
Look at the expression within the parentheses, [tex]\(3 - 7x^3\)[/tex]. This expression cannot be factored further using real numbers since it does not have any common factors or obvious patterns (like a difference of squares or cubes) that can be factored.
Thus, the complete factorization of the expression [tex]\(30x - 70x^4\)[/tex] is:
[tex]\[
-10x(7x^3 - 3)
\][/tex]
Note: The factors can also be expressed as [tex]\(10x(-7x^3 + 3)\)[/tex] which is equivalent to [tex]\(-10x(7x^3 - 3)\)[/tex]. Both forms are valid and correctly factor the original expression completely.
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