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Chapter 5: Problem 59

Factor completely. $$ 34 t^{3}-6 t $$

Short Answer

Expert verified
The factored form is 2t(17t^2 - 3).

Step by step solution

01

- Identify the Greatest Common Factor

Look at each term in the polynomial. Both terms, 34t^3 and -6t, have a common factor of 2t. Therefore, 2t is the greatest common factor.
02

- Factor Out the Common Factor

Factor out 2t from each term in the expression. This will leave us with 2t(17t^2 - 3).
03

- Verify the Factoring

Distribute 2t to ensure the correctness of the factoring: 2t(17t^2 - 3) = 34t^3 - 6t.

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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 a key concept in factoring polynomials. It is the highest factor that divides each term of the polynomial without leaving a remainder. To identify the GCF, we look at both the coefficients and the variables of each term.
In the given polynomial, 34t^3 and -6t have common factors. We notice that the number 2 can divide both 34 and -6. Additionally, both terms have a t. Thus, the GCF is 2t.
  • Example: For the terms 34t^3 and -6t, the GCF is 2t as it can be factored from both terms without any remainder.
  • Another way to find the GCF is to list out the factors of each term and find the highest common one.
Identifying the GCF is the first step in simplifying polynomial expressions.
polynomial expressions
Polynomial expressions are mathematical phrases that can contain numbers, variables, and the operations of addition, subtraction, and multiplication. They do not include division by a variable.
For example, the expression 34t^3 - 6t is a polynomial because it involves variables raised to a power, and each term is separated by either a plus or minus sign.
  • Terms: Parts of a polynomial that are added or subtracted. In 34t^3 - 6t, the terms are 34t^3 and -6t.
  • Coefficients: The numerical part of the terms. Here, they are 34 and -6.
  • Variables: The letters that represent numbers. Here, it is t.
Understanding polynomial expressions helps in recognizing how to factor them effectively.
factoring techniques
Factoring techniques are strategies used to simplify polynomial expressions by breaking them down into products of simpler expressions. One common technique is factoring out the greatest common factor (GCF).
Consider the polynomial 34t^3 - 6t. Here鈥檚 a step-by-step outline of the technique:
  • Identify the GCF: As found earlier, the GCF of 34t^3 and -6t is 2t.
  • Factor out the GCF: Use the GCF to factor each term in the polynomial: 34t^3 = 2t * 17t^2 and -6t = 2t * -3.
  • Rewrite the expression: Combine the factored terms: 2t(17t^2 - 3).
  • Verify the factoring: Distribute to check: 2t(17t^2 - 3) = 34t^3 - 6t.
Using these techniques makes complex polynomials easier to work with, facilitating further operations such as solving equations or simplifying expressions.

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