Problem 15 - College Algebra

Question

For the following exercises, factor by grouping. $$ 2 p^{2}-5 p-7 $$

Step-by-Step Solution

Answer

The factored expression is \((2p - 7)(p + 1)\).

Step 1: Recognize Grouping

The given quadratic expression is \(2p^2 - 5p - 7\). To factor this expression by grouping, we must split the middle term \(-5p\) into two terms whose coefficients multiply to the product of the coefficient of \(p^2\) term (which is 2) and the constant term (which is -7). This product is \(2 \times -7 = -14\). We need two numbers that multiply to -14 and add up to -5.

Step 2: Find Suitable Numbers

The numbers that multiply to -14 and add up to -5 are -7 and 2. Thus, we split the middle term \(-5p\) into \(-7p + 2p\).

Step 3: Rewrite the Expression

Rewrite the quadratic expression by substituting \(-5p\) with \(-7p + 2p\). The expression now becomes: \(2p^2 - 7p + 2p - 7\).

Step 4: Group Terms

Group the terms in pairs: \((2p^2 - 7p) + (2p - 7)\).

Step 5: Factor Common Terms in Groups

In the first group \((2p^2 - 7p)\), factor out a \(p\), resulting in \(p(2p - 7)\). In the second group \((2p - 7)\), there is already a common factor \(1\), which results in \(1(2p - 7)\).

Step 6: Factor the Entire Expression

Since both groups produce a common binomial factor \((2p - 7)\), factor out \((2p - 7)\) from the entire expression: \((2p - 7)(p + 1)\).

Key Concepts

Factoring by Grouping

Factoring by grouping is a highly effective technique for simplifying some polynomial expressions. This method involves rearranging and combining terms to reveal a common factor. It's particularly useful for quadratics where direct factoring isn't immediately evident. Begin by identifying an expression that can be split into groups that share a common factor. For example, in our exercise, we have the trinomial quadratic expression \(2p^2 - 5p - 7\). Notice that breaking down the middle term into two terms can help in arranging groups.

• First, determine the product of the leading coefficient of \(p^2\) and the constant term. This product guides how you group the terms.
• Find two numbers that both add up to the middle term and multiply to the product previously calculated.
• Re-structure the quadratic expression by replacing the middle term and group the terms into two pairs.
• Factor out a common term from each group. This will often reveal a shared binomial factor.
• Once you've isolated the common factor, factor it out to simplify the expression.

By systematically applying these steps, factoring by grouping simplifies expressions and prepares them for further algebraic manipulation.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are fundamental components in algebra. The expression given in the exercise, \(2p^2 - 5p - 7\), is a quadratic polynomial since its highest power of the variable is 2.

Understanding the structure of polynomials is essential. Here are some key concepts about polynomial expressions:

• Terms: A polynomial is made up of terms. Each term consists of a variable raised to an exponent and a coefficient. In this exercise, the terms are \(2p^2\), \(-5p\), and \(-7\).
• Degree: The degree of the polynomial is the highest exponent of its variable. For \(2p^2 - 5p - 7\), the degree is 2.
• Factoring: It involves rewriting the polynomial as a product of its factors. This can make solving equations and simplifying expressions easier.

Polynomials are extremely versatile and play a critical role in many areas of mathematics, from algebraic manipulation to calculus and beyond. Grasping their structure helps to apply different algebraic techniques effectively.

Algebraic Techniques

Algebraic techniques are essential tools in solving and simplifying mathematical problems. They include a variety of methods that assist in manipulating and understanding expressions and equations.

Let’s explore some key techniques used in factoring the given expression:

• Identifying and Splitting Terms: Breaking down complex terms into simpler parts allows for easier manipulation of the expression. In our exercise, we split the term \(-5p\) into \(-7p + 2p\), making it easier to group and factor.
• Finding a Common Factor: Look for common multiples in grouped terms. This technique simplifies the polynomial. By factoring out a common factor, as we did with \(p(2p - 7)\) and \(1(2p - 7)\), the polynomial can be rewritten more simply.
• Rewriting Expressions: Once terms are factored and common factors identified, the expression can often further be simplified or solved by rewriting it into a product of simpler expressions, as seen in the final solution \((2p - 7)(p + 1)\).

These techniques are not only crucial in algebra but also foster problem-solving skills useful across many disciplines requiring logical reasoning and manipulation of mathematical relationships.