Problem 68 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$p^{2}-7 p q-12 q^{2}$$

Step-by-Step Solution

Answer

The given quadratic expression, \(p^2 - 7pq - 12q^2\), can be factored completely as \((p - 4q)(p + 3q)\).

Step 1: Identify the General Form of the Quadratic Expression

The given quadratic expression is \(p^2 - 7pq - 12q^2\). The general form of a quadratic expression is similar to \(ax^2 + bx + c\), but in our case, we have variables p and q instead of x.

Step 2: Check for the Greatest Common Factor (GCF)

In our given expression \(p^2 - 7pq - 12q^2\), there are no common factors among the three terms. Therefore, we cannot factor out a GCF.

Step 3: Factor the Quadratic Expression

Since we cannot factor out any common factors, we'll proceed with factoring the quadratic expression. This means finding two binomials that multiply together to yield the original expression. We need to find factors for each term in the expression such that: 1. The product of the first terms in the binomials equals \(p^2\). 2. The product of the last term in the binomials equals \(-12q^2\). 3. The sum of the outer and inner terms in the binomials equals the middle term \(-7pq\). With this in mind: For \(p^2\), the only factors possible are p and p (since it's a square). For \(-12q^2\), let's list down all possible factor pairs: \((1q, -12q)\), \((-1q, 12q)\), \((2q, -6q)\), \((-2q, 6q)\), \((3q, -4q)\), \((-3q, 4q)\) The pair \((3q, -4q)\) works, as the sum of the outer and inner products equals the middle term: \(p(3q) + (-4q)p = -7pq\).

Step 4: Write the Final Factored Form

Now that we've found the pairs, we can write the final factored form of the quadratic expression: \((p - 4q)(p + 3q)\) The given quadratic expression, \(p^2 - 7pq - 12q^2\), can be factored completely as \((p - 4q)(p + 3q)\).

Key Concepts

greatest common factor

The greatest common factor (GCF) is the largest number or variable that divides each term in an expression without leaving a remainder. Knowing how to find the GCF is crucial when simplifying expressions or preparing to factor them completely. In the quadratic expression \(p^2 - 7pq - 12q^2\), the GCF would be any number or variable common to \(p^2\), \(-7pq\), and \(-12q^2\).

• First term: \(p^2\) consists of \(p\) repeated twice.
• Second term: \(-7pq\) includes \(p\) and \(q\), along with the constant \(-7\).
• Third term: \(-12q^2\) involves \(q\) squared with the constant \(-12\).

Upon examination, we can see that there are no common constants or variables across all three terms in this expression. Thus, there is no GCF, and we move on to find other means of factoring. Understanding this step is essential because factoring out a GCF simplifies future steps.

binomial multiplication

Binomial multiplication is a method used to multiply two binomials, which are expressions consisting of two terms. The result of binomial multiplication can inform us about how to reverse the process, which means factoring to find the original binomials from a quadratic expression.
When multiplying two binomials like \((a + b)(c + d)\), you use the distribution method:

• Multiply the first terms: \(a \cdot c\)
• Multiply the outer terms: \(a \cdot d\)
• Multiply the inner terms: \(b \cdot c\)
• Multiply the last terms: \(b \cdot d\)

Combine these products to give the final expression: \(ac + ad + bc + bd\). For our example, \((p - 4q)(p + 3q)\), applying binomial multiplication:
- First: \(p \cdot p = p^2\)
- Outer: \(p \cdot 3q = 3pq\)
- Inner: \(-4q \cdot p = -4pq\)
- Last: \(-4q \cdot 3q = -12q^2\)
Combine to form: \(p^2 + 3pq - 4pq - 12q^2\) which simplifies to \(p^2 - 7pq - 12q^2\). Recognizing these steps in reverse helps us factor quadratics.

quadratic expression factoring

Quadratic expression factoring involves breaking down a quadratic, typically in the form \(ax^2 + bx + c\), into a product of two binomials. This process not only helps solve quadratic equations but also simplifies expressions for further algebraic manipulation.
When factoring \(p^2 - 7pq - 12q^2\), we look for two binomials \((p + m)(p + n)\) that multiply to the quadratic. Here's the process:

• Ensure the first terms \(p \cdot p = p^2\)
• Find two numbers \(m\) and \(n\) that multiply to \(-12q^2\) (the constant term)
• These numbers \(m\) and \(n\) must also add to \(-7pq\) (the middle term)

Exploring factor pairs of \(-12q^2\):
- Consider \((3q, -4q)\) as this pair multiplies to \(-12q^2\) and their sum \(3pq - 4pq\) equals the middle term \(-7pq\)
Thus, the expression factors to \((p - 4q)(p + 3q)\). This technique of factoring quadratics helps solve a wide array of algebraic problems efficiently.