Problem 69 - Beginning and Intermediate Algebra

Question

Factor completely. $$3 x^{2}(y+6)^{2}-11 x(y+6)^{2}-20(y+6)^{2}$$

Step-by-Step Solution

Answer

The completely factored expression is: \((y+6)^2(x - 5)(3x + 4)\).

Step 1: Identify the GCF

The greatest common factor (GCF) of the given expression is \((y+6)^2\), as it is the common factor of all three terms: \(3x^2(y+6)^2 - 11x(y+6)^2 - 20(y+6)^2\)

Step 2: Factor out the GCF

Now, we will factor out \((y+6)^2\) from each term: \((y+6)^2(3x^2 - 11x - 20)\)

Step 3: Factor the quadratic expression

Now we need to further factor the quadratic expression inside the parentheses: \(3x^2 - 11x - 20\) To factor this quadratic expression, we can use the method of factoring by grouping. We will first find two numbers that multiply to the product of the first and last terms (-60) and add to the middle term (-11). The two numbers are -15 and 4. So we can rewrite the quadratic expression as: \(3x^2 - 15x + 4x - 20\) Now, group the terms and factor them: \(3x(x - 5) + 4(x - 5)\) Now we see that \((x-5)\) is a common factor in the two terms. Factor out \((x-5)\): \((x - 5)(3x + 4)\)

Step 4: Write the final factored expression

Now we can substitute the factored quadratic expression back into our main expression: \((y+6)^2(x - 5)(3x + 4)\) So, the given expression factored completely is: \((y+6)^2(x - 5)(3x + 4)\)

Key Concepts

Greatest Common Factor

When dealing with polynomial expressions, it's crucial to find the Greatest Common Factor (GCF). The GCF is the largest factor that divides each term of the polynomial. Identifying the GCF aids in simplifying the problem and is your first step toward factorization. In the given expression, \(3x^2(y+6)^2 - 11x(y+6)^2 - 20(y+6)^2\), the common factor across all terms is \((y+6)^2\). Recognizing \((y+6)^2\) as the GCF allows us to factor it out, providing a simpler equation to work with. It's like peeling off the outer layer, which uncovers the rest of the expression for further simplification.

Factoring by Grouping

After extracting the GCF, the expression \((y+6)^2(3x^2 - 11x - 20)\) needs further simplification. To tackle such a quadratic expression, factoring by grouping is an efficient strategy. This method involves organizing terms into groups and finding any common factors among them. Here, in the quadratic \(3x^2 - 11x - 20\), find two numbers that multiply to the product of the coefficient of the first term and the last term (i.e., \(3 \cdot -20 = -60\)) and that add up to the middle coefficient, \(-11\). The numbers \(-15\) and \(4\) achieve this. We rewrite the expression as:

• \(3x^2 - 15x + 4x - 20\)

Grouping like terms, we factor each set:

• \(3x(x - 5) + 4(x - 5)\)

Factor \((x - 5)\) out from each group to get:

• \((x - 5)(3x + 4)\)

Factoring by grouping reveals the hidden symmetry or common factors between the different groups, simplifying your polynomial further.

Quadratic Expression

A quadratic expression is any polynomial where the highest degree is 2, commonly in the form \(ax^2 + bx + c\). For the expression \(3x^2 - 11x - 20\), it's crucial to understand its structure to factor it properly. This expression is a quadratic because its highest exponent is 2. The process of factoring involves breaking it down into simpler multiplicative components, which essentially reverses the steps of expansion (such as foiling). Factoring quadratics often requires thinking backward: asking what pairs of terms or numbers combine through multiplication and addition to reconstruct the original expression. Identifying this allows us to split the expression and factor more manageable pieces, leading to eventual simplification.

Complete Factorization

After factoring both the GCF and the quadratic expression, you'll achieve what's known as complete factorization. This means expressing the polynomial as a product of its simplest factors, none of which can be broken down further. Returning to our example, we began with the expression \(3x^2(y+6)^2 - 11x(y+6)^2 - 20(y+6)^2\) and factored it to:

• \((y+6)^2(x - 5)(3x + 4)\)

The expression is now fully factored, indicated by the absence of any further common factors. Complete factorization is essential for solving higher-level mathematics problems, providing clear insight into the properties and potential solutions of the equation.