Factor completely: $12g^3 + 15g^2 - 28g - 35$
Understand the Problem
The question asks us to factor the given polynomial completely. We will need to find common factors within the expression to simplify it into a product of simpler polynomials.
Answer
$(4g + 5)(3g^2 - 7)$
Answer for screen readers
$(4g + 5)(3g^2 - 7)$
Steps to Solve
- Group the terms
We will group the first two terms and the last two terms together. $$ (12g^3 + 15g^2) + (-28g - 35) $$
- Factor out the greatest common factor (GCF) from each group
From the first group, the GCF is $3g^2$. Factoring this out, we have $3g^2(4g + 5)$. From the second group, the GCF is $-7$. Factoring this out, we have $-7(4g + 5)$. So we have: $$ 3g^2(4g + 5) - 7(4g + 5) $$
- Factor out the common binomial factor
We notice that $(4g + 5)$ is a common factor in both terms. We factor this out: $$ (4g + 5)(3g^2 - 7) $$
- Check for further factorization
The term $3g^2 - 7$ cannot be factored further using integer coefficients. Therefore, the complete factorization is: $$ (4g + 5)(3g^2 - 7) $$
$(4g + 5)(3g^2 - 7)$
More Information
Factoring by grouping is a useful technique when dealing with polynomials with four or more terms. It allows us to simplify the expression by finding common factors in pairs of terms.
Tips
A common mistake is to forget to factor out the negative sign in the second group, which would result in an incorrect common binomial factor. Another mistake is to stop factoring before the expression is completely factored.
AI-generated content may contain errors. Please verify critical information
