Problem 61 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$p^{3} q-17 p^{2} q^{2}+70 p q^{3}$$

Step-by-Step Solution

Answer

The completely factored form of the given expression \(\$p^3q - 17p^2q^2 + 70pq^3\$ is: \(pq(p - 2q)(p - 35q)\).

Step 1: Identify the GCF of the expression

We begin by looking for any common factors in all three terms of the given expression. The expression is: \(p^3q - 17p^2q^2 + 70pq^3\). The common factors are p and q. So, the GCF is pq.

Step 2: Factor out the GCF and rewrite the expression

Now we need to factor out the GCF (pq) from each term in the expression: \(p^3q - 17p^2q^2 + 70pq^3 = pq(p^2 - 17pq + 70q^2)\)

Step 3: Identify if the remaining expression can be factored further

Now let's check if the remaining quadratic expression inside the parenthesis can be factored further: \(p^2 - 17pq + 70q^2\) To factor this quadratic, we need to find two factors of 70 (constant term) that add up to -17 (coefficient of the linear term). These two factors are -2 and -35.

Step 4: Factor the remaining quadratic expression

Replace the linear term in the quadratic expression with the two factors we found in the previous step and then factor by grouping: \((p^2 - 2pq - 35pq + 70q^2)\) Now, group the terms: \((p^2 - 2pq) + (-35pq + 70q^2)\)

Step 5: Factor each group separately

Now, factor a GCF out of each group: \(p(p - 2q) - 35q(p - 2q)\)

Step 6: Factor out the common binomial factor

We can now factor out the common binomial factor \((p - 2q)\): \((p - 2q)(p - 35q)\)

Step 7: Rewrite the complete factored expression

Finally, we'll rewrite the factored expression by incorporating the factored GCF we found in step 2, getting the completely factored expression: \(pq(p^2 - 17pq + 70q^2) = pq(p - 2q)(p - 35q)\) The completely factored form of the given expression is: \(pq(p - 2q)(p - 35q)\).

Key Concepts

Greatest Common Factor (GCF)

Finding the greatest common factor (GCF) is the first step in factoring any expression, whether simple or complex. The GCF is the largest factor that all terms in an expression share. Identifying it can greatly simplify the factoring process by making the expression smaller and more manageable.
When approaching the expression \(p^3q - 17p^2q^2 + 70pq^3\), the focus is on finding the highest powers of variables and any numerical factors common to all terms. Here, both \(p\) and \(q\) appear in each term, making \(pq\) the GCF. By factoring out \(pq\), you simplify the expression to \(pq(p^2 - 17pq + 70q^2)\).
This first step is crucial as it reduces the complexity of the expression and paves the way for further factoring. Remember, pulling out the GCF can help you see patterns in the expression that were not initially clear.

Quadratic Expressions

Quadratic expressions are special algebraic expressions of the form \(ax^2 + bx + c\). The challenge with these expressions is factoring them, especially when they don't follow simple patterns like perfect squares or difference of squares.
In our exercise, the expression inside the parentheses after taking out the GCF, \(p^2 - 17pq + 70q^2\), is a quadratic expression in \(p\) and \(q\). To factor this, you need to focus on both the constant term, 70, and the linear coefficient, -17. The key is to find two numbers that multiply to give the constant term, 70, and add up to the coefficient of the linear term, -17.
For \(p^2 - 17pq + 70q^2\), these numbers are -2 and -35, because -2 times -35 equals 70, and -2 plus -35 equals -17. By finding these two numbers, you can write the quadratic expression as a product of two binomials, making the overall expression easier to factor.

Factoring by Grouping

Factoring by grouping is a method used when an expression doesn't initially look like it can be factored directly. It works by rearranging terms so that they can be grouped in pairs, each of which has a common factor.
In the example \(p^2 - 17pq + 70q^2\), you identified factors -2 and -35 that help you split the middle term. Break the expression into two pairs: \(p^2 - 2pq\) and \(-35pq + 70q^2\).
The next step is to factor each group separately. For \(p^2 - 2pq\), factor out \(p\), which gives \(p(p - 2q)\). For the second group \(-35pq + 70q^2\), factor out \(-35q\), yielding \(-35q(p - 2q)\).
With factoring by grouping, you've revealed a common binomial \((p - 2q)\), which can be factored out of the entire expression, resulting in \((p - 2q)(p - 35q)\). This method simplifies complex expressions into a product of polynomials that are easier to work with.