Problem 3 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(2x\).

Step 2: 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 \(2x(-x^2 + 8)\).

Key Concepts

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.