Problem 74 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$(m-n) v^{2}+11(m-n) v+30(m-n)$$

Step-by-Step Solution

Answer

The completely factored expression of the given polynomial, \((m-n)v^2+11(m-n)v+30(m-n)\), is \((m-n)(v + 5)(v + 6)\).

Step 1: Identify the GCF

In this given expression, \((m-n)v^2+11(m-n)v+30(m-n)\), we need to find the greatest common factor amongst all three terms. We can see that the factor \((m-n)\) is present in every term.

Step 2: Factor out the GCF

Since \((m-n)\) is the GCF, we can factor it out from every term: \((m-n)(v^2+11v+30)\) Now, we have a quadratic expression inside the parentheses, \(v^2+11v+30\).

Step 3: Factor the quadratic expression

In order to factor the quadratic expression, we need to find two numbers whose product equals the constant term (30) and whose sum equals the coefficient of the linear term (11). These two numbers are 5 and 6, since: Product: \(5 \times 6 = 30\) Sum: \(5 + 6 = 11\) So, we can rewrite the quadratic expression as: \(v^2+11v+30 = (v + 5)(v + 6)\)

Step 4: Write the completely factored expression

Replace the quadratic expression inside the parentheses with its factored form, resulting in the completely factored expression: \((m-n)(v^2+11v+30) = (m-n)(v + 5)(v + 6)\) The completely factored expression is \((m-n)(v + 5)(v + 6)\).

Key Concepts

Understanding the Greatest Common Factor (GCF)

When you're working on factoring polynomials, finding the Greatest Common Factor (GCF) is a crucial first step. The GCF is the largest factor that can evenly divide all terms in the expression. In the equation \((m-n)v^2+11(m-n)v+30(m-n)\), notice how \((m-n)\) appears in every term. This makes \((m-n)\) our GCF.

You can think of the GCF as the 'biggest piece' that all the terms share. By pulling it out, you're simplifying the expression. This step helps reveal what lies underneath, making it easier to work with the remaining parts.

• Look for common terms or numbers in all parts of the expression.
• Factor these common parts out and write them outside a bracket.
• This should make the remaining expression simpler to handle.

Exploring Quadratic Expressions

A quadratic expression is any expression that includes a variable squared, known as the quadratic term. In the reduced expression \(v^2+11v+30\), \(v^2\) is the quadratic term, 11v is the linear term, and 30 is the constant term. Quadratic expressions are common in algebra and can often be factored into simpler expressions.

Factoring a quadratic expression involves finding two numbers that multiply to give the constant term while adding up to the coefficient of the linear term.

• Quadratic expressions usually follow the format \(ax^2+bx+c\).
• Use the numbers that satisfy both the product and sum requirements to split the middle term.
• This approach will simplify the quadratic, making it straightforward to factor.

Mastering Factoring Techniques

Factoring techniques allow us to break down complex expressions into simpler, multiplied components. After factoring out the GCF, like we've done to obtain \((m-n)(v^2+11v+30)\), the next step is to focus on quadratic expressions.

To factor \(v^2+11v+30\), you need to find two numbers whose product is 30 and whose sum is 11. Here, 5 and 6 are the numbers that work, resulting in \((v + 5)(v + 6)\).

Here’s a simple approach:

• Identify pairs that multiply to the constant term.
• Check which pair adds up to the coefficient of \(v\).
• Rewrite the quadratic by splitting the middle term accordingly.

Notice how these techniques simplify the polynomial, as seen in \((m-n)(v + 5)(v + 6)\). These methods apply across various expressions, reinforcing skills for solving more complicated algebraic equations.