Factor completely: $15x^3 - 5x^2 - 33x + 11$
Understand the Problem
The question asks us to completely factor the polynomial expression $15x^3 - 5x^2 - 33x + 11$. This means we need to find expressions that, when multiplied together, give us the original polynomial.
Answer
$(3x - 1)(5x^2 - 11)$
Answer for screen readers
$(3x - 1)(5x^2 - 11)$
Steps to Solve
- Factor by grouping
Group the terms in pairs: $(15x^3 - 5x^2) + (-33x + 11)$
- Factor out the greatest common factor (GCF) from each pair
From the first pair, the GCF is $5x^2$. From the second pair, the GCF is $-11$: $5x^2(3x - 1) - 11(3x - 1)$
- Factor out the common binomial factor
Notice that both terms now have a common factor of $(3x - 1)$. Factor this out: $(3x - 1)(5x^2 - 11)$
- Final factored form
The expression is now completely factored
$(3x - 1)(5x^2 - 11)$
More Information
The factored form of the polynomial $15x^3 - 5x^2 - 33x + 11$ is $(3x - 1)(5x^2 - 11)$.
Tips
A common mistake is not factoring out the negative sign from the second group, which would result in an incorrect common binomial factor. Also, one might stop after factoring out the GCF from each pair and not recognize the common binomial factor.
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