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Chapter 0: Problem 3

1鈥�8 ? Factor out the common factor. $$ -2 x^{3}+16 x $$

Short Answer

Expert verified
The expression factors to \(2x(-x^2 + 8)\).

Step by step solution

01

Identify the common factors

Examine the terms in \(-2x^3 + 16x\). Both terms contain the factor \(x\), and their coefficients \(-2\) and \(16\) share the greatest common divisor, which is \(2\). Therefore, the common factor of both terms is \(2x\).
02

Factor out the common factor

Factor out \(2x\) from each term of the expression. This involves dividing each term by \(2x\). - From \(-2x^3\), we get \(-2x^3 \div 2x = -x^2\).- From \(16x\), we get \(16x \div 2x = 8\).Thus, the expression becomes \(2x(-x^2 + 8)\).

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Key Concepts

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

Common Factor
When working with algebraic expressions, a common factor is a number or variable that divides exactly into each term of the expression. Identifying common factors is a fundamental step in simplifying algebraic expressions, particularly in the context of polynomial factoring.

In the expression \(-2x^3 + 16x\), both terms include the variable \(x\). Further, the coefficients, namely \(-2\) for \(-2x^3\) and \(16\) for \(16x\), have a common divisor, which is \(2\).
  • Coefficient \(-2\) and \(16\) share \(2\) as their greatest common factor.
  • Variable \(x\) is present in both terms.
Identifying these shared elements allows us to factor out \(2x\) from the original expression, which is the process of rewriting it in a simpler form where the common factor is outside a parenthesis.
Polynomial Factoring
Polynomial factoring involves rewriting a polynomial (an expression composed of variables and coefficients) as a product of simpler polynomials. This process helps in solving equations, simplifying expressions, and uncovering root values in algebra.

Here, the expression \(-2x^3 + 16x\) is simplified by factoring. Identify and factor out the common factor \(2x\):
  • Divide each term by \(2x\):
  • \(-2x^3\) divided by \(2x\) equals \(-x^2\).
  • \(16x\) divided by \(2x\) equals \(8\).
Once factored, the expression transforms into: \[ 2x(-x^2 + 8) \]This conversion relies heavily on understanding each term's components and applying the factors correctly. Factoring polynomials eases many algebraic processes, including finding solutions to equations and graphing functions.
Greatest Common Divisor
The Greatest Common Divisor (GCD) of two or more numbers is the largest number that divides each of them without leaving a remainder. Finding the GCD is crucial in simplifying algebraic expressions by allowing us to determine the greatest factor each term shares.

In our example, \(-2x^3 + 16x\), we focus first on the coefficients \(-2\) and \(16\) to find their GCD.
  • The factors of \(-2\) are \(1\) and \(2\).
  • The factors of \(16\) are \(1, 2, 4, 8,\) and \(16\).
  • \(2\) is the largest number common to both sets of factors.
Therefore, the GCD for the coefficients is \(2\). Combined with the shared variable \(x\), this results in the common factor \(2x\), which is factored out of the expression. Finding the GCD is essential as it helps simplify expressions efficiently by reducing like terms to their most basic forms.

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Most popular questions from this chapter

Number of Molecules A sealed room in a hospital, measuring 5 \(\mathrm{m}\) wide, 10 \(\mathrm{m}\) long, and 3 \(\mathrm{m}\) high, is filled with pure oxygen. One cubic meter contains 1000 \(\mathrm{L}\) , and 22.4 \(\mathrm{L}\) of any gas contains \(6.02 \times 10^{23}\) molecules (Avogadro's number). How many molecules of oxygen are there in the room?

Simplify the expression and eliminate any negative exponent(s). $$ \left(3 y^{2}\right)\left(4 y^{5}\right) $$

Simplify the expression and eliminate any negative exponent(s). $$ \left(\frac{q^{-1} r s^{-2}}{r^{-5} s q^{-8}}\right)^{-1} $$

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer correct to the number of significant digits indicated by the given data. $$ \left(1.062 \times 10^{24}\right)\left(8.61 \times 10^{19}\right) $$

\(89-96\) m State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ \frac{2}{4+x}=\frac{1}{2}+\frac{2}{x} $$
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