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Chapter 6: Problem 45

Write an equivalent expression by factoring out the greatest common factor. $$ 2 t^{2}+8 t $$

Short Answer

Expert verified
Factor out the GCF to get \(2(t^2 + 4t)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the terms in the expression. The terms in our expression are \(2t^2\) and \(8t\). The GCF of 2 and 8 is 2.
02

Factor Out the GCF

Next, factor out the GCF from each term in the expression. Divide each term by 2 and write the expression as a product of the GCF and the remaining factors. \[2t^2 + 8t = 2(t^2) + 2(4t) = 2(t^2 + 4t)\]
03

Verify the Factored Expression

To verify the factored expression, multiply the GCF by each term inside the parentheses and check if it simplifies back to the original expression. \[2(t^2 + 4t) = 2t^2 + 8t\]

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Key Concepts

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

Greatest Common Factor (GCF)
To solve an algebraic expression by factoring, we first need to understand the greatest common factor, or GCF. The GCF is the largest number that can divide all terms of an expression without leaving a remainder. This helps simplify the expression by reducing it to a more manageable form. For example, in the expression \(2t^2 + 8t\), the terms are \(2t^2\) and \(8t\). Identify the GCF by looking at the coefficients and the variables. Here, the highest number that can divide both 2 and 8 is 2. We also notice both terms have at least one 't'. However, in factoring out the GCF, we use only the numerical factor, which is 2. So, the GCF is just 2.
Polynomial Factoring
Factoring polynomials involves expressing the polynomial as a product of simpler polynomials. Once you've identified the GCF as explained, the next step is to factor it out from each term. In our example, the initial expression is \(2t^2 + 8t\). Since the GCF is 2, you factor out 2 from each term. This gives us the following:

\[2t^2 + 8t = 2(t^2) + 2(4t) = 2(t^2 + 4t)\]

This step helps simplify the polynomial into a product involving the GCF and another polynomial \(t^2 + 4t\), which is easier to work with for further operations.
Algebraic Simplification
Algebraic simplification is the process of reducing complex expressions to their simplest form. After factoring out the GCF, it鈥檚 crucial to verify the expression is correct. Simplification verifies that the factored form returns to the original expression when multiplied back.

Let's check:
  • Take the factored expression: \(2(t^2 + 4t)\)
  • Distribute the 2 back into each term inside the parentheses: \[2(t^2) + 2(4t) = 2t^2 + 8t\]
This confirms that the factored expression is equivalent to the original. Simplification is key to solving and verifying algebraic expressions efficiently.

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Most popular questions from this chapter

Factor: $$ 5 x^{5}-5 x^{4}+x^{3}-x^{2}+3 x-3 $$

Factor completely. Assume that variables in exponents represent positive integers. $$ a^{2}+2 a b+b^{2}-c^{2}+6 c-9 $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{3}+3 x^{2}+x-1=0 $$

Solve. Round any irrational solutions to the nearest thousandth. $$ r^{2}+16=8 r $$

Write an equivalent expression by factoring. Assume that all exponents are natural numbers. $$ 3 a^{n+1}+6 a^{n}-15 a^{n+2} $$
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