Problem 55 - Introductory Algebra for College Students

Question

Factor completely. $$2 r^{3}+8 r^{2}-64 r$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(2 r^{3}+8 r^{2}-64 r\) is \(2r(r - 4)(r + 8)\).

Step 1: Factor out the Greatest Common Factor

From the given polynomial \(2 r^{3}+8 r^{2}-64 r\), factor out the greatest common factor '2r' from each term to get \(2r(r^{2} + 4r - 32)\)

Step 2: Factor the Quadratic Expression

Factor the quadratic expression \(r^{2} + 4r - 32\) by finding two numbers that add up to 4 (the coefficient of 'r') and multiply to -32 (the constant term). After factoring the quadratic expression, we get \((r - 4)(r + 8)\).

Step 3: Write Final Result

Now, substitute back the common factor factorized in first step. The completely factorized form of the polynomial is \(2r(r - 4)(r + 8)\).

Key Concepts

Greatest Common Factor

When dealing with polynomials, one of the first things to do is identify the Greatest Common Factor (GCF) among the terms.
This is a number or expression that divides each term in the polynomial without leaving a remainder.
It's like the biggest piece you can "take out" from each part of the expression without breaking any mathematical rules.

• To find the GCF, look at both the numerical coefficients and the variables in each term.
• For example, in the polynomial \(2r^3 + 8r^2 - 64r\), each term has a factor of \(2r\).
• This means \(2r\) is the GCF.

Once you identify the GCF, you can pull it out and write it in front of a set of parentheses.
Inside the parentheses, you will write the simplified form of the polynomial where each term has been divided by the GCF.
This step simplifies the polynomial and makes it easier to factor further.

Quadratic Expressions

A quadratic expression is a type of polynomial with a degree of 2, meaning the highest power of the variable is squared.
Quadratic expressions often follow the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
Factorizing a quadratic expression involves finding two binomials that multiply together to give the original quadratic.

• In the expression \(r^2 + 4r - 32\), we need to think of two numbers that multiply to \(-32\) and add to \(4\).
• These numbers are \(-4\) and \(8\).

So, the quadratic can be factorized as \((r - 4)(r + 8)\).
Recognizing patterns such as these makes the factorization process much more manageable.

Polynomial Factoring Steps

Factoring polynomials, especially those involving both cubic terms and quadratics, generally follows a structured series of steps.
Understanding these steps can streamline the process and help avoid mistakes.

• Step 1: Identify the Greatest Common Factor (GCF) and factor it out from the polynomial.
• Step 2: Focus on the remaining expression, which usually simplifies to a quadratic or another manageable form.
• Step 3: Factor the quadratic expression using methods such as splitting the middle term or using a factorization formula.
• Step 4: Combine the GCF with the factorized expression to write the polynomial in its completely factorized form.

In our example, we applied these steps to factor \(2r^3 + 8r^2 - 64r\) into \(2r(r - 4)(r + 8)\).
Practicing these steps will improve your skills and confidence in polynomial factorization.