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Chapter 6: Problem 76

Factor completely. If a polynomial is prime, state this. $$ t^{4}+7 t^{2}-3 t^{3}-21 t $$

Short Answer

Expert verified
(t - 3) t (t^{2} + 7)

Step by step solution

01

- Group the polynomial terms

Group the polynomial into pairs: \( t^{4} - 3t^{3} \) and \( 7t^{2} - 21t \).
02

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

For the first group \( t^{4} - 3t^{3} \), the GCF is \( t^{3} \), giving \( t^{3}(t - 3) \). For the second group \( 7t^{2} - 21t \), the GCF is \( 7t \), giving \( 7t(t - 3) \).
03

Factor out the common binomial factor

Both groups now contain the common factor \( (t - 3) \). Factor \( (t - 3) \) out of both terms: \( t^{3}(t - 3) + 7t(t - 3) \) becomes \( (t - 3)(t^{3} + 7t) \).
04

Factor further if possible

Evaluate \( t^{3} + 7t \) for further factoring. Factor out \( t \), giving \( t(t^{2} + 7) \).
05

Write the complete factorization

Combine the factors: \( (t - 3) (t) (t^{2} + 7) \). The polynomial is now completely factored.

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Key Concepts

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

Grouping Method
The grouping method is a useful technique for factoring polynomials with four or more terms. The main idea is to organize the polynomial into smaller groups and factor out the greatest common factors (GCF) within each group. This simplifies the polynomial and makes it easier to identify common factors.
In the given problem, the polynomial is grouped as follows:
\ \ - \ \ First group: \( t^{4} - 3t^{3} \). \ - Second group: \( 7t^{2} - 21t \).\ By factoring each group separately, we can make the polynomial easier to handle and ultimately simplify it completely.
Greatest Common Factor (GCF)
The greatest common factor (GCF) of a set of terms is the largest factor that each term in the set shares. Finding and factoring out the GCF is a crucial step in simplifying polynomials. Here's how it works in this exercise:
For the first group \(t^{4} - 3t^{3}\), the GCF is \(t^{3}\). When we factor out \(t^{3}\), we get \(t^{3}(t - 3)\).
For the second group \(7t^{2} - 21t\), the GCF is \(7t\). When we factor out \(7t\), we get \(7t(t - 3)\).
By doing this for each group, we can now focus on the common binomial factor \((t - 3)\).
Binomial Factor
A binomial factor is an expression with two terms that is obtained by factoring out a common factor from a polynomial. In our exercise, after grouping and factoring each segment, we found that both groups share a common binomial factor. This is a crucial recognition step in the polynomial factoring process.
The binomial factor here is \((t - 3)\). We factored this out from both groups: \(t^{3}(t - 3) + 7t(t - 3)\).
To finalize, we factor \((t - 3)\) from each group, resulting in \((t - 3)(t^{3} + 7t)\).
Finally, we can factor further: \(t^{3} + 7t\) becomes \(t(t^{2} + 7)\).
Combining everything, the complete factorization is \((t - 3)(t)(t^{2} + 7)\).

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