To factor the polynomial [tex]\(8x^3 - 125\)[/tex] with integer coefficients, follow this step-by-step process:
1. Recognize the form of the polynomial: The polynomial [tex]\(8x^3 - 125\)[/tex] is a difference of cubes. The difference of cubes can be expressed in the general form [tex]\(a^3 - b^3\)[/tex], which factors into [tex]\((a - b)(a^2 + ab + b^2)\)[/tex].
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]: In the given polynomial, [tex]\(8x^3\)[/tex] and [tex]\(125\)[/tex] are both perfect cubes. We can write:
[tex]\[
8x^3 = (2x)^3 \quad \text{and} \quad 125 = 5^3
\][/tex]
So, we have [tex]\(a = 2x\)[/tex] and [tex]\(b = 5\)[/tex].
3. Apply the difference of cubes formula: Now, substitute [tex]\(a = 2x\)[/tex] and [tex]\(b = 5\)[/tex] into the formula [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex].
4. Substitute and simplify:
[tex]\[
8x^3 - 125 = (2x)^3 - 5^3
\][/tex]
Applying the difference of cubes formula:
[tex]\[
(2x - 5)\left((2x)^2 + (2x)(5) + 5^2\right)
\][/tex]
5. Simplify inside the parentheses:
[tex]\[
(2x - 5)\left(4x^2 + 10x + 25\right)
\][/tex]
Thus, the factored form of the polynomial [tex]\(8x^3 - 125\)[/tex] is:
[tex]\[
(2x - 5)(4x^2 + 10x + 25)
\][/tex]
So the final answer is:
[tex]\[
(2x - 5)(4x^2 + 10x + 25)
\][/tex]
