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Answer :
We want to factor the polynomial
[tex]$$15x^3 - 25x^2 + 75x - 125.$$[/tex]
Step 1. Factor out the greatest common factor
Notice that every coefficient is divisible by 5, so we start by factoring out 5:
[tex]$$
15x^3 - 25x^2 + 75x - 125 = 5\left(3x^3 - 5x^2 + 15x - 25\right).
$$[/tex]
Step 2. Factor the cubic polynomial
Next, we factor the cubic expression inside the parentheses:
[tex]$$
3x^3 - 5x^2 + 15x - 25.
$$[/tex]
Through factorization, we find that this cubic can be written as a product of a linear factor and a quadratic factor:
[tex]$$
3x^3 - 5x^2 + 15x - 25 = (3x - 5)(x^2 + 5).
$$[/tex]
You can verify this by expanding the factors:
[tex]\[
\begin{aligned}
(3x - 5)(x^2 + 5) &= 3x \cdot x^2 + 3x \cdot 5 - 5 \cdot x^2 - 5 \cdot 5 \\
&= 3x^3 + 15x - 5x^2 - 25 \\
&= 3x^3 - 5x^2 + 15x - 25.
\end{aligned}
\][/tex]
Step 3. Write the complete factorization
Substitute back into the expression with the common factor:
[tex]$$
15x^3 - 25x^2 + 75x - 125 = 5(3x - 5)(x^2 + 5).
$$[/tex]
Thus, the fully factored form of the polynomial is:
[tex]$$
\boxed{5(3x - 5)(x^2 + 5)}.
$$[/tex]
[tex]$$15x^3 - 25x^2 + 75x - 125.$$[/tex]
Step 1. Factor out the greatest common factor
Notice that every coefficient is divisible by 5, so we start by factoring out 5:
[tex]$$
15x^3 - 25x^2 + 75x - 125 = 5\left(3x^3 - 5x^2 + 15x - 25\right).
$$[/tex]
Step 2. Factor the cubic polynomial
Next, we factor the cubic expression inside the parentheses:
[tex]$$
3x^3 - 5x^2 + 15x - 25.
$$[/tex]
Through factorization, we find that this cubic can be written as a product of a linear factor and a quadratic factor:
[tex]$$
3x^3 - 5x^2 + 15x - 25 = (3x - 5)(x^2 + 5).
$$[/tex]
You can verify this by expanding the factors:
[tex]\[
\begin{aligned}
(3x - 5)(x^2 + 5) &= 3x \cdot x^2 + 3x \cdot 5 - 5 \cdot x^2 - 5 \cdot 5 \\
&= 3x^3 + 15x - 5x^2 - 25 \\
&= 3x^3 - 5x^2 + 15x - 25.
\end{aligned}
\][/tex]
Step 3. Write the complete factorization
Substitute back into the expression with the common factor:
[tex]$$
15x^3 - 25x^2 + 75x - 125 = 5(3x - 5)(x^2 + 5).
$$[/tex]
Thus, the fully factored form of the polynomial is:
[tex]$$
\boxed{5(3x - 5)(x^2 + 5)}.
$$[/tex]
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