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Chapter 5: Problem 92

Write an equivalent expression by factoring. $$2 x^{3 a}+8 x^{a}+4 x^{2 a}$$

Short Answer

Expert verified
2x^a (x^{2a} + 2x^a + 4)

Step by step solution

01

Identify Common Factors

Look at all the terms in the expression and identify the common factor. In this case, each term contains a factor of 2 and a factor of some power of x.
02

Factor Out the Greatest Common Factor

The greatest common factor (GCF) is the highest factor that can divide all terms in the polynomial. For the given expression, the GCF is 2 and \(x^a\) because each term contains both of these factors. Factoring out \(2x^a\), we get: \[ 2x^a (x^{2a} + 4 + 2x^a) \]
03

Simplify the Expression Inside the Parentheses

Simplify the expression inside the parentheses to get the final factored form: \[ 2x^a (x^{2a} + 2x^a + 4) \]

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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 the largest number or expression that can exactly divide all terms in a polynomial.
Finding the GCF requires identifying common factors in the terms of the polynomial.
These factors can be coefficients (numbers) or variables with exponents.
In our example, let's consider the polynomial: 2x^{3a} + 8x^{a} + 4x^{2a}. All terms contain a factor of 2 and some power of x.
Emphasizing extraction of every common factor simplifies the polynomial.To find the GCF for the polynomial:
  • List the highest common coefficient, which is 2.
  • List the common variable factor, in this case, it is x, with the smallest exponent, which is a.

Combining these, we see that the GCF is 2x^a.
This is the factor we will factor out from the polynomial.
factoring expressions
Factoring expressions involves breaking down a polynomial into simpler terms that, when multiplied together, give you the original polynomial.
This step is significant as it simplifies polynomial expressions, making it easier to solve or simplify further operations.
Now consider the polynomial we have: 2x^{3a} + 8x^{a} + 4x^{2a}.
We know from the GCF section that we want to factor out 2x^a. This means rewriting the polynomial like this:
2x^a( ?)We need to determine what goes into the parentheses.
  • Divide each original term by the GCF:
    (2x^{3a})/(2x^a) = x^{2a}
    (8x^{a})/(2x^a) = 4
    (4x^{2a})/(2x^a) = 2x^a
Putting all together, we have: 2x^a (x^{2a} + 4 + 2x^a).
This is the factored form of the polynomial.
polynomial simplification
Polynomial simplification is the process of transforming a polynomial into its simplest form. This often involves factoring out common terms and combining like terms.
In our given expression, we finally reached a form that looks simpler and easy to work with.
To recapitulate, we started with: 2x^{3a} + 8x^{a} + 4x^{2a}
and factored out the GCF 2x^a, which gave us: 2x^a (x^{2a} + 4 + 2x^a).
Notice how the polynomial inside the parentheses is now much cleaner.
The goal of polynomial simplification is to make the expression less complex while retaining its original value.
This cleaned-up form can then be used in further calculations or to find specific solutions.

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Most popular questions from this chapter

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