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Chapter 6: Problem 30

Factor completely, or state that the polynomial is prime. $$2 x^{3}-8 x$$

Short Answer

Expert verified
The factored form of the polynomial \(2x^{3}-8x\) is \(2x(x-2)(x+2)\).

Step by step solution

01

Identify the GCF

The greatest common factor between the terms \(2x^{3}\) and \(-8x\) is \(2x\). This is because \(2x\) is the largest factor that divides both terms evenly.
02

Divide the Entire Polynomial by the GCF

Once the GCF has been identified, divide each term in the polynomial by this GCF. So \(2x^{3}\) divided by \(2x\) is \(x^{2}\) and \(-8x\) divided by \(2x\) is \(-4\).
03

Write Down the Factored Form

The polynomial, after factoring, becomes the GCF times the result from the previous step, i.e. \(2x(x^{2}-4)\). However, note that \(x^{2}-4\) is a difference of squares, which can be further factored into \((x-2)(x+2)\). So a more complete factorization is \(2x(x-2)(x+2)\).

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Key Concepts

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

Greatest Common Factor (GCF)
Factoring a polynomial often begins by identifying the Greatest Common Factor (GCF) among its terms. The GCF is the largest factor that divides each term evenly. In our example, the polynomial is \(2x^3 - 8x\). To determine the GCF:
  • Look at the coefficients (the numbers directly in front of the variables). For \(2x^3\) and \(-8x\), the coefficients are 2 and -8. The largest number that divides both is 2.
  • Examine the variable part. The variable in both terms is \(x\). Since \(x^3\) and \(x\) both have \(x\), we take the lowest power, which is \(x\).
The GCF for the polynomial is \(2x\), meaning each term can be divided by \(2x\) without leaving a remainder. Factoring out the GCF simplifies the polynomial and prepares it for further factorization.
Difference of Squares
The concept of a difference of squares comes into play when recognizing patterns within a polynomial. This happens when you have an expression like \(a^2 - b^2\). The difference of squares can be rewritten as the product of two binomials:
  • The formula is \(a^2 - b^2 = (a - b)(a + b)\).
In our example, after factoring out the GCF, we found the expression \(x^2 - 4\). Notice \(x^2 - 4\) can be expressed as a difference of squares:
  • Here, \(a^2\) is \(x^2\) and \(b^2\) is 4, which is also \(2^2\).
So, \(x^2 - 4\) becomes \((x - 2)(x + 2)\). This technique simplifies expressions and is crucial for achieving a complete factorization of a polynomial.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler, easier-to-handle expressions, which multiply together to give the original polynomial. This process makes solving equations and simplifying expressions more manageable.
In our exercise, the polynomial \(2x^3 - 8x\) undergoes factorization as follows:
  • Start by identifying the GCF, which is \(2x\).
  • Divide the entire polynomial by \(2x\), leaving us with \(x^2 - 4\).
  • Recognize \(x^2 - 4\) as a difference of squares and factor it into \((x - 2)(x + 2)\).
The completely factored form of the polynomial, now expressed as \(2x(x - 2)(x + 2)\), is made of linear factors. Each factor represents a simpler expression that builds up to the original polynomial when multiplied. Understanding and applying these steps ensures easier manipulation and comprehension of polynomials.

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Most popular questions from this chapter

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$12 y^{2}-11 y+2$$

The polynomial \(4 x^{2}+100\) is the sum of two squares and therefore cannot be factored. If the general factoring strategy is used to factor a polynomial, at least two factorizations are necessary before the given polynomial is factored completely.

Factor completely. $$(x+5)^{2}-20(x+5)+100$$

Exercises 150鈥�152 will help you prepare for the material covered in the next section. Evaluate \((3 x-1)(x+2)\) for \(x=\frac{1}{3}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(x^{2}-9=(x-3)^{2}\) for any real number \(x\)
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