Problem 142 - Elementary Algebra

Question

Factor completely. \(3 u^{2}-12 u-36\)

Step-by-Step Solution

Answer

3(u - 6)(u + 2)

Step 1: Identify the Greatest Common Factor (GCF)

Look for the greatest common factor (GCF) in the polynomial terms. Observe that all terms in the polynomial share a common factor of 3.

Step 2: Factor Out the GCF

Factor out the GCF from each term of the polynomial: \[ 3u^{2} - 12u - 36 = 3(u^{2} - 4u - 12) \]

Step 3: Factor the Quadratic Expression

Now, focus on factoring the quadratic expression inside the parentheses. Identify two numbers that multiply to -12 (the constant term) and add to -4 (the coefficient of the middle term): These numbers are -6 and 2. This allows the quadratic expression to be rewritten and factored: \[u^{2} - 4u - 12 = (u - 6)(u + 2) \]

Step 4: Combine All Factors

Combine the factors from step 2 and step 3: \[ 3(u^{2} - 4u - 12) = 3(u - 6)(u + 2) \]

Key Concepts

greatest common factor

The first step in tackling polynomial factorization is to find the Greatest Common Factor (GCF). The GCF of a list of terms is the largest factor that divides each of them without leaving a remainder. For the polynomial \(3u^2 - 12u - 36\), we need to find the common factor in the coefficients 3, -12, and -36. In this case, the GCF is 3. By factoring out the GCF, we simplify the polynomial, making it \(3(u^2 - 4u - 12)\). This step helps in making further factorization easier. Identifying the GCF is a critical process in algebraic factorization because it reduces the polynomial to more manageable terms.

quadratic expressions

A quadratic expression is a polynomial of the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are coefficients. Our simplified polynomial \(u^2 - 4u - 12\) is a quadratic expression. Factoring quadratic expressions involves finding two binomials \((x + m)(x + n)\) such that their product equals the original quadratic. For \(u^2 - 4u - 12\), we need two numbers whose product is \(-12\) (the constant term) and whose sum is \(-4\) (the coefficient of the middle term). In this case, these numbers are \(-6\) and \(2\). This allows us to write the quadratic expression as \((u - 6)(u + 2)\). Now, we have successfully factored the quadratic expression into two binomials.

algebraic factorization

Algebraic factorization is the process of breaking down a polynomial into a product of simpler polynomials. The objective is to write the given polynomial as a product of its factors. Incorporating both the GCF and factoring the quadratic expression, our polynomial transforms into \(3(u - 6)(u + 2)\). This final product represents the completely factored form of \(3u^2 - 12u - 36\). Each term in the factorized form is a building block of the original polynomial. The benefits of factorizing polynomials include simplifying geometry problems, solving equations, and understanding polynomial behavior at different values of the variable. Factorization plays a pivotal role in algebra because it helps reveal the roots and other properties of polynomials.