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Chapter 5: Problem 57

Factor. $$ t^{8}+5 t^{7}-14 t^{6} $$

Short Answer

Expert verified
The factored form is \(t^6 (t + 7)(t - 2)\).

Step by step solution

01

- Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) for all the terms. Here, all terms share the factor of \(t^6\). The GCF is \(t^6\).
02

- Factor Out the GCF

Factor out the GCF \(t^6\) from each term. This transforms the expression: \[t^{8}+5t^{7}-14t^{6} = t^6 (t^2 + 5t - 14)\]
03

- Factor the Quadratic Expression

Next, factor the quadratic expression \(t^2 + 5t - 14\). We need to find two numbers that multiply to \(-14\) and add up to \(5\). These numbers are \(7\) and \(-2\). The quadratic can be factored as: \[t^2 + 5t - 14 = (t + 7)(t - 2)\]
04

- Combine All Factors

Finally, combine the factors from the previous steps: \[t^8 + 5t^7 - 14t^6 = t^6 (t + 7)(t - 2)\]

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is one of the foundational tools for simplifying polynomials. To find the GCF, look at all the terms in the polynomial and identify the largest factor that is common to each term. For the expression \(t^8 + 5t^7 - 14t^6\), each term includes a power of \(t\).
The smallest exponent in these terms is 6, making \(t^6\) the GCF. When you factor out the GCF, you are simplifying the expression by pulling out common elements.
This helps to make the problem easier to handle and often reveals simpler polynomial forms that can be further factored.
Quadratic Expressions
A quadratic expression is a polynomial of degree two and typically takes the form \(ax^2 + bx + c\). In our problem, after factoring out the GCF \(t^6\), the remaining expression was \(t^2 + 5t - 14\), which is quadratic.
The goal is to factor this into two binomials. You look for two numbers that multiply to the constant term (\(-14\) in this case) and add up to the linear coefficient (\(5\)).
For this example, these numbers are \(7\) and \(-2\), as \(7 \times (-2) = -14\) and \(7 + (-2) = 5\). This makes our expression: \(t^2 + 5t - 14 = (t + 7)(t - 2)\).
Quadratic expressions are central in algebra and recognizing their structure helps in many areas of mathematics.
Algebraic Factorization
Algebraic factorization involves breaking down a complex polynomial into simpler factors that, when multiplied together, give the original polynomial.
The process often starts with identifying the GCF. After that, other techniques like factoring quadratic expressions or special polynomial forms can be applied.
The given problem started with \(t^8 + 5t^7 - 14t^6\).
After factoring out the GCF \(t^6\), we had \(t^6 (t^2 + 5t - 14)\), which was then factored further into \(t^6 (t + 7)(t - 2)\).
Algebraic factorization is a crucial skill in algebra as it simplifies polynomials, making it easier to solve equations and understand the polynomial's behavior.

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