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Chapter 5: Problem 60

Factor completely. $$ 13 t^{3}-26 t $$

Short Answer

Expert verified
The fully factored form is 13t(t^2 - 2).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Examine the terms in the expression. Both terms, 13t^3 and -26t, have a common factor. The GCF is 13t.
02

Factor Out the GCF

Factor out the 13t from each term: 13t(t^2 - 2)
03

Verify the Factored Form

Multiply the factored terms back out to ensure the expression is equivalent to the original expression: 13t(t^2 - 2) gives 13t^3 - 26t, confirming our factorization is correct.

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Key Concepts

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

Greatest Common Factor
When working with polynomial expressions, one of the first steps often involves finding the Greatest Common Factor (GCF). The GCF is the highest number or variable that can evenly divide all terms in the polynomial.

For example, in the expression \(13t^3 - 26t\), we need to look at each term. The term \(13t^3\) contains factors of 13 and t, while the term \(-26t\) contains factors of 26 and t. Since 13 is a factor of both 13 and 26, and both terms have at least one 't', the GCF here is \(13t\).

Using the GCF simplifies the expression, making further operations easier and often leading to a factored form that is easier to interpret.
Factoring
Factoring is the process of breaking down an expression into a product of other expressions, which when multiplied together give the original expression.

In our example, after identifying the GCF, the next step is to factor it out. Given \(13t\) as the GCF of \(13t^3 - 26t\), we rewrite the original expression as the product of \(13t\) and another polynomial:

\[ 13t^3 - 26t = 13t(t^2 - 2) \]

Here, we've factored the expression into \(13t\), which is multiplied by \(t^2 - 2\).
This process makes the expression more manageable and reveals its structure more clearly.
Polynomial Expressions
Polynomial expressions consist of terms made up of coefficients and variables raised to integer powers.

In the expression \(13t^3 - 26t\), there are two terms: \(13t^3\) (a cubic term) and \(-26t\) (a linear term).

Understanding the structure of polynomial expressions is crucial when performing operations such as factoring.

Each term has a coefficient (13 and -26) and a variable factor (t^3 and t, respectively).
Recognizing polynomials and their components helps in identifying the GCF and in the process of simplifying or factoring expressions.

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Most popular questions from this chapter

Mary Louise is attempting to solve \(x^{3}+20 x^{2}+4 x+80=0\) with a graphing calculator. Unfortunately, when she graphs \(y_{1}=x^{3}+20 x^{2}+4 x+80\) in a standard \([-10,10,-10,10]\) window, she sees no graph at all, let alone any \(x\) -intercept. Can this problem be solved graphically? If so, how? If not, why not?

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