Problem 70 - Elementary and Intermediate Algebra

Question

Factor. If an expression is prime, so indicate. $$ 20 y^{2}-93 y-35 $$

Step-by-Step Solution

Answer

The expression factors to \((20y + 7)(y - 5)\).

Step 1: Identify Coefficients

For the quadratic expression \(20y^2 - 93y - 35\), identify the coefficients: \(a = 20\), \(b = -93\), and \(c = -35\).

Step 2: Calculate the Product \(a \cdot c\)

Multiply the leading coefficient \(a\) and the constant term \(c\): \(20 \times -35 = -700\).

Step 3: Find Two Numbers that Multiply to \(a \cdot c\) and Add to \(b\)

We need two numbers that multiply to \(-700\) and add to \(-93\). These numbers are \(-100\) and \(7\) because \(-100 \times 7 = -700\) and \(-100 + 7 = -93\).

Step 4: Split the Middle Term Using the Two Numbers Found

Rewrite the middle term \(-93y\) as \(-100y + 7y\). The expression becomes \(20y^2 - 100y + 7y - 35\).

Step 5: Group Terms for Factoring by Grouping

Group the terms into two pairs: \((20y^2 - 100y) + (7y - 35)\).

Step 6: Factor Each Group Separately

Factor out the greatest common factor from each group. From the first group \(20y^2 - 100y\), factor out \(20y\) to get \(20y(y - 5)\). From the second group \(7y - 35\), factor out \(7\) to get \(7(y - 5)\).

Step 7: Factor by Grouping

Since \(y - 5\) is a common factor, factor it out: \((20y + 7)(y - 5)\).

Step 8: Verify the Factorization

Expand \((20y + 7)(y - 5)\) using the distributive property to ensure it equals the original expression. \((20y + 7)(y - 5) = 20y^2 - 100y + 7y - 35 = 20y^2 - 93y - 35\). The factorization is correct.

Key Concepts

Quadratic Equation

A quadratic equation is an algebraic expression of the second degree, which means it includes a term with the variable squared. It is typically written in the standard form as \(ax^2 + bx + c = 0\), where:

• \(a\), \(b\), and \(c\) are constants.
• \(x\) represents the variable or unknown.
• The term \(ax^2\) is the leading term, with \(a\) as the leading coefficient.

In our example of factoring, the expression \(20y^2 - 93y - 35\) follows the pattern of a quadratic equation where:

• \(a = 20\)
• \(b = -93\)
• \(c = -35\)

Quadratic equations can have real or complex solutions, and factoring is just one of the methods used to solve or simplify them. The goal of factoring is to rewrite the quadratic equation as a product of two binomials, making it easier to solve for the variable.

Factoring by Grouping

Factoring by grouping is a valuable technique used to factor more complex algebraic expressions, such as certain quadratic expressions. This method is particularly useful when the expression is not easily factorable through basic inspection. The process involves several steps:

• Splitting the middle term: In our exercise, we found numbers whose product was equal to the product of \(a\) and \(c\) and whose sum equaled \(b\). Here, we split \(-93y\) into two terms: \(-100y\) and \(+7y\).
• Creating groups: Next, we grouped the expression into two pairs, resulting in \((20y^2 - 100y) + (7y - 35)\).
• Factoring each group: We then looked for the greatest common factor (GCF) in each group. For \(20y^2 - 100y\), the GCF was \(20y\); for \(7y - 35\), it was \(7\).
• Factoring the expression: With a common factor identified in both groups, i.e., \(y - 5\), we factored further to get \((20y + 7)(y - 5)\).

This method ensures that even seemingly complicated expressions can be broken down into manageable parts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operators, forming a core component of algebra. They represent relationships between quantities and can be as simple as a single number or variable, or as complex as the quadratic expression we are factoring.Some essential features of algebraic expressions include:

• Terms: Parts of the expression separated by addition or subtraction, such as \(20y^2\), \(-93y\), and \(-35\) in our exercise.
• Coefficients: Numbers that multiply the variables. For example, the coefficient of \(y^2\) in our expression is 20.
• Variables: Symbols that represent unknown values. In our quadratic expression, the variable is \(y\).
• Constants: Numbers on their own without variables, like \(-35\) in the expression.

Algebraic expressions enable us to model real-world situations in a mathematical format, allowing for easier manipulation and solution of problems. Through different techniques like factoring by grouping, these expressions can be simplified further for better understanding and application.