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Chapter 6: Problem 86

Factor by grouping, if possible, and check. $$ t^{3}+6 t^{2}-2 t-12 $$

Short Answer

Expert verified
(t^{3}+6 t^{2}-2 t-12) = (t^2 - 2)(t + 6)

Step by step solution

01

Group the terms

Start by grouping the terms into pairs: equation: t^{3} + 6t^{2} - 2t - 12 = (t^{3} + 6t^{2}) + (-2t - 12)
02

Factor out the greatest common factor (GCF) from each group

For the first group Equations: (t^{3} + 6t^{2}), factor out a GCF of t虏: (t^2)(t+6). For the second group Equation: (-2t -12), factor out a GCF of -2: -2(t+6).
03

Factor out the common binomial factor

Notice each group contains a common binomial factor (t+6): t^{3}+6 t^{2}-2 t-12 = (t^2 - 2)(t + 6)
04

Verify the factorization

Multiply the factors back to ensure they equal the original expression: (t^2 - 2)(t + 6) = t^{3} + 6 t^{2} - 2 t - 12. Since they match, the factorization is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is the process of breaking down an expression into a product of simpler expressions. It helps simplify complex polynomials and solve equations. For example, when factoring the polynomial \( t^{3}+6 t^{2}-2 t-12 \), we aim to express it as a product of two binomials or a binomial and a more straightforward polynomial.
Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) of a group of terms is the largest factor that divides all terms without leaving a remainder. GCF is crucial in simplifying expressions.
For instance, in our example, we grouped terms: \( t^{3} + 6t^{2} - 2t - 12 \) and identified the GCF for each group:
  • For the first group, \( t^{3} + 6t^{2} \), the GCF is \( t^{2} \). This results in \( t^2(t + 6) \).
  • For the second group, \( -2t - 12 \), the GCF is \( -2 \). This simplifies to \( -2(t + 6) \).
This step ensures the complexity of the original polynomial is reduced in manageable terms.
Polynomial Factorization
Polynomial factorization is the process of expressing a polynomial as a product of its factors.
In our example, once we factored the groups, we saw a common binomial factor in both groups: \( (t + 6) \). We can then factor this common binomial out, yielding:\[ t^{3}+6 t^{2}-2 t-12 = (t^2 - 2)(t + 6) \].
This format simplifies the polynomial into a product of simpler factors. Factoring by grouping is especially handy for polynomials with more than one variable.
Verification
Verification confirms our factorization is correct. We multiply the factors back to see if the result matches the original polynomial.
For example, multiplying the factors of our polynomial (t^2 - 2)(t + 6):
  • First, expand: \[ t^{3} + 6t^{2} - 2t - 12 \].
  • This matches our original polynomial.
If the expanded form matches the original, the factorization is verified.
Verification helps avoid mistakes and ensure our factorization is accurate.

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