Problem 83 - A Graphical Approach to College Algebra

Question

Factor by any method. $$4 p^{2}+3 p-1$$

Step-by-Step Solution

Answer

The expression factors to \((4p - 1)(p + 1)\).

Step 1: Identify the Type of Quadratic Expression

The given expression is a quadratic trinomial in the form \( ax^2 + bx + c \), where \( a = 4 \), \( b = 3 \), and \( c = -1 \). Our goal is to factor this trinomial into two binomials.

Step 2: Determine Factor Pairs of \( a \cdot c \)

Calculate \( a \cdot c \): \( 4 \times (-1) = -4 \). We need to find two numbers whose product is \(-4\) and whose sum is \(3\). The pair \(4\) and \(-1\) satisfies this because \(4 + (-1) = 3\).

Step 3: Rewrite the Middle Term

Use the pair found to split the middle term: \(4p^2 + 4p - p - 1\). This decomposition helps in factoring by grouping.

Step 4: Factor by Grouping

Group the terms: \((4p^2 + 4p) + (-p - 1)\). Factor out the greatest common factor in each group: \(4p(p + 1) - 1(p + 1)\).

Step 5: Factor Out the Common Factor

Since \((p + 1)\) is a common factor, extract it: \((4p - 1)(p + 1)\). This is the factored form of the original expression.

Key Concepts

Understanding Trinomials

A trinomial is an algebraic expression composed of exactly three terms. Typically, in algebra, we encounter trinomials within quadratic expressions. A standard form of a quadratic trinomial looks like this: \( ax^2 + bx + c \), where:

• \( a \), \( b \), and \( c \) are coefficients.
• \( x \) is the variable.

These expressions are essential because they often represent parabolas when graphed. Solving or factoring these expressions helps in finding roots or x-intercepts of the parabola.
In our exercise, the trinomial given is \( 4p^2 + 3p - 1 \). It fits the pattern \( ax^2 + bx + c \) with \( a = 4 \), \( b = 3 \), and \( c = -1 \). Understanding this structure is the stepping stone to using methods like factoring by grouping.

Factoring by Grouping Explained

Factoring by grouping is a systematic technique used primarily to simplify expressions. It is particularly useful for factoring trinomials when straightforward factorization doesn't quickly present itself.
To factor by grouping, follow these general steps:

• First, multiply the first and last coefficients \( a \) and \( c \).
• Then, find a pair of numbers that multiply to this product and add up to the middle coefficient \( b \).
• Use these numbers to rewrite or decompose the middle term \( bx \).
• Finally, group the terms into two pairs and factor out the greatest common factor (GCF) from each pair.

For example, in the trinomial \( 4p^2 + 3p - 1 \), after calculating the product of \( 4 \) and \( -1 \) resulting in \( -4 \), we found the pair \( 4 \) and \( -1 \) suited to split \( 3p \) into \( 4p - p \). This strategy allowed us to group and factor as \((4p^2 + 4p) + (-p - 1)\), leading eventually to the factors \((4p - 1)(p + 1)\).

Quadratic Expressions and Their Significance

Quadratic expressions form a core part of algebra and are fundamental in mathematics. These are any polynomial expressions where the highest degree (or power) of the variable is squared.
Quadratics typically have the form \( ax^2 + bx + c \) and when factored, represent products of binomials. The ability to factor these expressions accurately is vital because:

• It simplifies complex calculations.
• It aids in solving quadratic equations.
• It helps in finding real and complex roots quickly.

In graphical terms, quadratic expressions describe parabolas. Factoring provides another description of parabolas in terms of their roots or x-intercepts, giving insight into their graphical behavior.
The exercise challenges us to factor \( 4p^2 + 3p - 1 \), which through methodical steps, reveals that the quadratic can be written as a product of \((4p - 1)(p + 1)\). This makes solving equations easier, offering insight that examining the unfactored expression might not directly show.