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Answer :
To factor the expression [tex]\(7x^5 - 21x^4 + 14x^3\)[/tex] completely, follow these steps:
1. Find the Greatest Common Factor (GCF):
Look at each term in the expression: [tex]\(7x^5\)[/tex], [tex]\(-21x^4\)[/tex], and [tex]\(14x^3\)[/tex]. Notice that each coefficient (7, -21, 14) shares a common factor of 7, and the smallest power of [tex]\(x\)[/tex] in the terms is [tex]\(x^3\)[/tex].
Therefore, the greatest common factor is [tex]\(7x^3\)[/tex].
2. Factor Out the GCF:
Divide each term by the GCF, [tex]\(7x^3\)[/tex], to factor it out of the expression.
[tex]\[
\begin{align*}
7x^5 &\div 7x^3 = x^2, \\
-21x^4 &\div 7x^3 = -3x, \\
14x^3 &\div 7x^3 = 2.
\end{align*}
\][/tex]
So, the expression inside the parentheses becomes [tex]\(x^2 - 3x + 2\)[/tex].
3. Write the Expression:
After factoring out the GCF, the expression becomes:
[tex]\[
7x^3(x^2 - 3x + 2)
\][/tex]
Now, the expression is factored completely. This means it's expressed as a product of the greatest common factor and a polynomial that cannot be factored further using integer coefficients.
1. Find the Greatest Common Factor (GCF):
Look at each term in the expression: [tex]\(7x^5\)[/tex], [tex]\(-21x^4\)[/tex], and [tex]\(14x^3\)[/tex]. Notice that each coefficient (7, -21, 14) shares a common factor of 7, and the smallest power of [tex]\(x\)[/tex] in the terms is [tex]\(x^3\)[/tex].
Therefore, the greatest common factor is [tex]\(7x^3\)[/tex].
2. Factor Out the GCF:
Divide each term by the GCF, [tex]\(7x^3\)[/tex], to factor it out of the expression.
[tex]\[
\begin{align*}
7x^5 &\div 7x^3 = x^2, \\
-21x^4 &\div 7x^3 = -3x, \\
14x^3 &\div 7x^3 = 2.
\end{align*}
\][/tex]
So, the expression inside the parentheses becomes [tex]\(x^2 - 3x + 2\)[/tex].
3. Write the Expression:
After factoring out the GCF, the expression becomes:
[tex]\[
7x^3(x^2 - 3x + 2)
\][/tex]
Now, the expression is factored completely. This means it's expressed as a product of the greatest common factor and a polynomial that cannot be factored further using integer coefficients.
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