Factor completely: $2q^3 + 3q^2 + 20q + 30$
Understand the Problem
The question asks to factor the given polynomial completely. The expression is $2q^3 + 3q^2 + 20q + 30$. Factoring involves expressing the polynomial as a product of simpler polynomials or factors.
Answer
$(2q + 3)(q^2 + 10)$
Answer for screen readers
$(2q + 3)(q^2 + 10)$
Steps to Solve
-
Factor by Grouping Group the terms in pairs: $$(2q^3 + 3q^2) + (20q + 30)$$
-
Factor out the Greatest Common Factor (GCF) from each group From the first group, the GCF is $q^2$. From the second group, the GCF is $10$. $$q^2(2q + 3) + 10(2q + 3)$$
-
Factor out the common binomial factor Notice that both terms now have a common factor of $(2q + 3)$. We can factor this out: $$(2q + 3)(q^2 + 10)$$
-
Check if further factoring is possible The term $(2q + 3)$ is linear and cannot be factored further. The term $(q^2 + 10)$ is a sum of squares, which cannot be factored further using real numbers.
$(2q + 3)(q^2 + 10)$
More Information
The polynomial $2q^3 + 3q^2 + 20q + 30$ factors into $(2q + 3)(q^2 + 10)$. This is the complete factorization over real numbers.
Tips
A common mistake is to stop after factoring out the GCF from each pair of terms and not factor out the common binomial factor. Also, one might try to factor $q^2+10$ further, but it's not factorable over real numbers.
AI-generated content may contain errors. Please verify critical information
