Factor completely: \(5u^5 - 10u^4 - 4u^3 + 8u^2\)
Understand the Problem
The question asks us to factor the polynomial expression (5u^5 - 10u^4 - 4u^3 + 8u^2) completely. We need to find the factors that, when multiplied together, give us the original expression.
Answer
$u^2(u - 2)(5u^2 - 4)$
Answer for screen readers
$u^2(u - 2)(5u^2 - 4)$
Steps to Solve
- Factor out the greatest common factor.
The greatest common factor (GCF) of the terms $5u^5$, $-10u^4$, $-4u^3$, and $8u^2$ is $u^2$. Factoring out $u^2$ from the expression, we get: $$ u^2(5u^3 - 10u^2 - 4u + 8) $$
- Factor by grouping.
Now, we factor the cubic polynomial $5u^3 - 10u^2 - 4u + 8$ by grouping. We group the first two terms and the last two terms: $$ (5u^3 - 10u^2) + (-4u + 8) $$ From the first group, we factor out $5u^2$, and from the second group, we factor out $-4$: $$ 5u^2(u - 2) - 4(u - 2) $$ Now, we have a common factor of $(u - 2)$, so we factor that out: $$ (u - 2)(5u^2 - 4) $$
- Recognize a difference of squares.
Observe that $5u^2 - 4$ can be written as $(\sqrt{5}u)^2 - (2)^2$, which is a difference of squares. We can factor it as: $$ (\sqrt{5}u - 2)(\sqrt{5}u + 2) $$ However, we can also treat $5u^2 - 4$ as not factorable over the integers.
- Write the fully factored expression.
Combining the factors, we have: $$ u^2(u - 2)(5u^2 - 4) $$
If factoring over the real numbers is accepted, we can also express the answer as: $$ u^2(u-2)(\sqrt{5}u - 2)(\sqrt{5}u + 2) $$
$u^2(u - 2)(5u^2 - 4)$
More Information
The factored form of the given polynomial is $u^2(u - 2)(5u^2 - 4)$. Further factoring can lead to $u^2(u-2)(\sqrt{5}u - 2)(\sqrt{5}u + 2)$, but the initial answer is generally accepted.
Tips
A common mistake is not factoring out the greatest common factor, $u^2$, in the first step. Another common mistake is incorrectly factoring the difference of squares, or not recognizing it at all. It is also possible to make sign errors when factoring by grouping.
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