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Chapter 2: Problem 114

Completely factor the expression over the real numbers. $$x^{3}-5 x^{2}-14 x$$

Short Answer

Expert verified
The completely factored form of the expression \(x^{3}-5x^{2}-14x\) over the real numbers is \(x(x-7)(x+2)\).

Step by step solution

01

Identify Common Factor

Look for any common factor in all terms of the polynomial. In this case, all terms in the polynomial share x as a common factor. So, we can factor out x.
02

Factor Out

Factor out the common factor from each term in the polynomial: \(x(x^{2}-5x-14)\). This leaves us with a quadratic factor inside parentheses.
03

Quadratic Factorization

Factorize the quadratic expression. We look for two numbers that multiply to -14 and add to -5. After some trial and error, or by using the quadratic formula, we can identify these numbers as -7 and 2. So, we can write the expression as: \(x(x-7)(x+2)\).

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Key Concepts

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

Common Factor
When factoring polynomials, finding a common factor is often the first step in simplifying the expression. A common factor is a term that is present in each term of the polynomial, serving as a shared multiple that can be factored out. Identifying a common factor simplifies the polynomial into a product of simpler expressions.

For example, in the polynomial \(x^3 - 5x^2 - 14x\), 'x' is the common factor because it appears in each term of the polynomial. Factoring out the 'x' gives us \(x(x^2 - 5x - 14)\). By doing this, we reduce the polynomial from a cubic to a quadratic expression within the parentheses, making it easier to manage and further factorize.
Quadratic Factorization
After factoring out the common factor from a polynomial, you may be left with a quadratic expression, which is an expression of the form \(ax^2 + bx + c\). The goal of quadratic factorization is to break this expression down into a product of two binomials.

To factor the quadratic expression \(x^2 - 5x - 14\), we search for two numbers that, when multiplied, give the constant term (in this case, -14) and when added, give the coefficient of the 'x' term (in this case, -5). Through examination or using methods such as the AC method or the quadratic formula, we find these numbers to be -7 and 2, thus allowing us to rewrite the quadratic as \(x - 7)(x + 2)\). This fully factors our original polynomial into \(x(x - 7)(x + 2)\).
Polynomial Expressions
A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A single term of a polynomial, called a monomial, can consist of constants, variables, and exponents. The degree of the polynomial indicates its highest power.

The given expression, \(x^3 - 5x^2 - 14x\), is a cubic polynomial (since its highest degree is 3) in one variable 'x'. Recognizing the structure of polynomial expressions is key to understanding how to factor them efficiently. Terms with the highest powers are generally tackled first by factoring out common factors or using quadratic factorization techniques when applicable to break the polynomial down into its simplest binomial or trinomial forms.

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Most popular questions from this chapter

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(B(t) \geq 900 .\) Explain what the solution of the inequality represents.

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$(x+2)^{2}<25$$

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(B(t) \geq W(t) .\) Explain what the solution of the inequality represents.

Completely factor the expression over the real numbers. $$5(x+5) x^{1 / 3}+4 x^{4 / 3}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=2 x+\frac{8}{x-5}-2$$
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