Factor completely: 36x^3 + 45x^2 - 28x - 35
Understand the Problem
The question asks us to factor the given cubic polynomial completely. This means we need to express the polynomial as a product of simpler polynomials or constants.
Answer
$(4x + 5)(9x^2 - 7)$
Answer for screen readers
$(4x + 5)(9x^2 - 7)$
Steps to Solve
- Factor by grouping
Group the first two terms and the last two terms together:
$(36x^3 + 45x^2) + (-28x - 35)$
- Factor out the greatest common factor (GCF) from each group
From the first group, the GCF is $9x^2$. Factoring this out, we get:
$9x^2(4x + 5)$
From the second group, the GCF is $-7$. Factoring this out, we get:
$-7(4x + 5)$
- Combine the factored terms
Now, we can rewrite the expression as:
$9x^2(4x + 5) - 7(4x + 5)$
- Factor out the common binomial factor
Notice that $(4x + 5)$ is a common factor. Factoring this out, we get
$(4x + 5)(9x^2 - 7)$
- Check if further factoring is possible
The term $(9x^2 - 7)$ can be expressed as the difference of squares, but since $7$ is not a perfect square, it cannot be factored further using integer coefficients.
Therefore, the completely factored form is:
$(4x + 5)(9x^2 - 7)$
$(4x + 5)(9x^2 - 7)$
More Information
Factoring by grouping is a useful technique when you have a polynomial with four terms. It involves grouping the terms in pairs, factoring out the GCF from each pair, and then looking for a common binomial factor.
Tips
A common mistake is to incorrectly identify the GCF when factoring out each group. Another mistake is to forget to factor out the negative sign when the third term is negative. Finally, some students may stop after factoring out the GCF from each pair and forget to factor out the common binomial factor.
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