Certainly! Let’s factor the polynomial step by step:
Given polynomial: [tex]\(500u^3v - 864v\)[/tex]
1. Factor out the common term:
The term [tex]\(v\)[/tex] is common in both terms of the polynomial.
[tex]\[
500u^3v - 864v = v(500u^3 - 864)
\][/tex]
2. Factor the remaining polynomial inside the parentheses:
Consider the polynomial [tex]\(500u^3 - 864\)[/tex].
3. Factor out the greatest common divisor of the coefficients 500 and 864:
The greatest common divisor of 500 and 864 is 4.
[tex]\[
500u^3 - 864 = 4(125u^3 - 216)
\][/tex]
So the polynomial becomes:
[tex]\[
v(500u^3 - 864) = v \cdot 4(125u^3 - 216) = 4v(125u^3 - 216)
\][/tex]
4. Factor the cubic polynomial [tex]\(125u^3 - 216\)[/tex]:
Recognize that [tex]\(125u^3 - 216\)[/tex] can be factored as a difference of cubes. Recall that a difference of cubes can be factored using the identity:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
Here, [tex]\(125u^3 = (5u)^3\)[/tex] and [tex]\(216 = 6^3\)[/tex], therefore, we apply the identity:
[tex]\[
125u^3 - 216 = (5u)^3 - 6^3 = (5u - 6)((5u)^2 + (5u)(6) + 6^2)
\][/tex]
Simplifying inside the parentheses,
[tex]\[
(5u)^2 = 25u^2,
\][/tex]
[tex]\[
(5u)(6) = 30u,
\][/tex]
[tex]\[
6^2 = 36.
\][/tex]
So, we have:
[tex]\[
(5u - 6)(25u^2 + 30u + 36)
\][/tex]
5. Combine all parts:
Putting it all together, we get:
[tex]\[
500u^3v - 864v = 4v(125u^3 - 216) = 4v(5u - 6)(25u^2 + 30u + 36)
\][/tex]
Hence, the fully factored form of the polynomial [tex]\(500u^3v - 864v\)[/tex] is:
[tex]\[
4v(5u - 6)(25u^2 + 30u + 36)
\][/tex]
