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Answer :
To factor the expression [tex]\(5x - 625x^4\)[/tex], let's go through it step-by-step:
1. Identify Common Factors:
Look for terms that can be factored out from both parts of the expression [tex]\(5x\)[/tex] and [tex]\(-625x^4\)[/tex]. Notice that both terms have a common factor of [tex]\(5x\)[/tex].
2. Factor Out the Common Factor:
Factor [tex]\(5x\)[/tex] out of each term:
[tex]\[
5x - 625x^4 = 5x(1 - 125x^3)
\][/tex]
3. Factor the Remaining Expression:
Now, let's focus on the expression inside the parentheses, [tex]\(1 - 125x^3\)[/tex]. Notice that this is a difference of cubes:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
In this case, [tex]\(a = 1\)[/tex] and [tex]\(b = 5x\)[/tex], since [tex]\((5x)^3 = 125x^3\)[/tex]. Apply the difference of cubes formula:
[tex]\[
1 - (5x)^3 = (1 - 5x)(1^2 + 1(5x) + (5x)^2)
\][/tex]
4. Simplify the Remaining Expression:
Simplify inside the factored form:
[tex]\[
1^2 + 1(5x) + (5x)^2 = 1 + 5x + 25x^2
\][/tex]
5. Combine All Parts:
Incorporate these results into the expression [tex]\(5x(1 - 125x^3)\)[/tex]:
[tex]\[
5x(1 - 125x^3) = 5x(1 - 5x)(1 + 5x + 25x^2)
\][/tex]
Thus, the factored form of [tex]\(5x - 625x^4\)[/tex] is:
[tex]\[
-5x(5x - 1)(25x^2 + 5x + 1)
\][/tex]
1. Identify Common Factors:
Look for terms that can be factored out from both parts of the expression [tex]\(5x\)[/tex] and [tex]\(-625x^4\)[/tex]. Notice that both terms have a common factor of [tex]\(5x\)[/tex].
2. Factor Out the Common Factor:
Factor [tex]\(5x\)[/tex] out of each term:
[tex]\[
5x - 625x^4 = 5x(1 - 125x^3)
\][/tex]
3. Factor the Remaining Expression:
Now, let's focus on the expression inside the parentheses, [tex]\(1 - 125x^3\)[/tex]. Notice that this is a difference of cubes:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
In this case, [tex]\(a = 1\)[/tex] and [tex]\(b = 5x\)[/tex], since [tex]\((5x)^3 = 125x^3\)[/tex]. Apply the difference of cubes formula:
[tex]\[
1 - (5x)^3 = (1 - 5x)(1^2 + 1(5x) + (5x)^2)
\][/tex]
4. Simplify the Remaining Expression:
Simplify inside the factored form:
[tex]\[
1^2 + 1(5x) + (5x)^2 = 1 + 5x + 25x^2
\][/tex]
5. Combine All Parts:
Incorporate these results into the expression [tex]\(5x(1 - 125x^3)\)[/tex]:
[tex]\[
5x(1 - 125x^3) = 5x(1 - 5x)(1 + 5x + 25x^2)
\][/tex]
Thus, the factored form of [tex]\(5x - 625x^4\)[/tex] is:
[tex]\[
-5x(5x - 1)(25x^2 + 5x + 1)
\][/tex]
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