Problem 108 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Factor \(-3 x^{3}+12 x^{2}+36 x\) completely.

Step-by-Step Solution

Answer

-3x(x-6)(x+2)

Step 1 - Identify the greatest common factor (GCF)

First, identify the greatest common factor (GCF) of the terms. The terms are -3x^3, 12x^2, and 36x.The GCF of these terms is 3x.

Step 2 - Factor out the GCF

Divide each term by the GCF (3x), factoring it out of the polynomial: -3x^3 + 12x^2 + 36x = 3x(-x^2 + 4x + 12)

Step 3 - Factor the quadratic expression

Next, factor the quadratic expression -x^2 + 4x + 12.Look for two numbers that multiply to -12 (the product of the coefficient of x^2 term and the constant term) and add to 4 (the coefficient of the x term). These numbers are -2 and 6.

Step 4 - Rewrite the quadratic expression

Rewrite the quadratic expression using the numbers -2 and 6:-x^2 + 4x + 12 = -(x^2 - 4x - 12) = -(x-6)(x+2)

Step 5 - Factor the entire expression

Combine all the factored parts together to write the complete factorized form: 3x(-x^2 + 4x + 12) = 3x[-(x-6)(x+2)] = -3x(x-6)(x+2)

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept in polynomial factorization. It's the highest number that can evenly divide all the terms in a polynomial. Identifying the GCF simplifies the polynomial and makes further factoring steps easier.
For example, in the polynomial \(-3x^3 + 12x^2 + 36x\), the GCF is the largest term that factors out from -3x^3, 12x^2, and 36x. Here, the GCF is 3x:

• -3x^3 ÷ 3x = -x^2
• 12x^2 ÷ 3x = 4x
• 36x ÷ 3x = 12

After factoring out the GCF, the expression simplifies to 3x(-x^2 + 4x + 12).

Quadratic Expression

A quadratic expression is a polynomial of degree 2, usually in the form ax^2 + bx + c. Factoring a quadratic expression involves breaking it down into simpler terms that, when multiplied, give back the original expression.
In our example, after factoring out the GCF, we have the quadratic expression inside the parentheses: -x^2 + 4x + 12. We need to rewrite this quadratic in a factored form.
The key is to find two numbers that multiply to the product of the coefficient of x^2 term (-1) and the constant term (12), which is -12, and add up to the coefficient of the x term (4).

• The numbers -2 and 6 fit these criteria
• -2 * 6 = -12 and -2 + 6 = 4

This allows us to rewrite the quadratic as -(x-6)(x+2).

Factoring

Factoring involves breaking down a complex expression into product of simpler expressions. It's used to simplify polynomials, solve quadratic equations, and find roots. Steps in factoring include:

• Identifying the GCF and factoring it out
• Rewriting the remaining polynomial in a factorable form
• Combining all factored parts into one expression

In our exercise, we already factored out the GCF (3x) and rewrote the quadratic expression (-x^2 + 4x + 12) as -(x-6)(x+2).
Now, combining these steps gives us:

• 3x(-x^2 + 4x + 12) = 3x[-(x-6)(x+2)]
• = -3x(x-6)(x+2)

Thus, the completely factored form of the original polynomial is \(-3x(x-6)(x+2)\).