Problem 80 - Algebra A Combined Function

Question

Perform each indicated operation. Explain the difference between the two expressions. a. \((8 x-3)-(5 x-2)\) b. \((8 x-3)(5 x-2)\)

Step-by-Step Solution

Answer

Expression (a) simplifies to \(3x - 1\); Expression (b) simplifies to \(40x^2 - 31x + 6\).

Step 1: Simplify Expression (a)

For the expression \((8x-3) - (5x-2)\),apply the distributive property to eliminate the parentheses: \[8x - 3 - 5x + 2\]Combine like terms by subtracting coefficients of \(x\): \[(8x - 5x) + (-3 + 2)\]This simplifies to \[3x - 1\].

Step 2: Simplify Expression (b)

For the expression \((8x-3)(5x-2)\), use the distributive property (also known as the FOIL method, for First, Outer, Inner, Last) to multiply each term in the first binomial by each term in the second binomial. Multiply each pair as follows: \[8x \times 5x = 40x^2\]\[8x \times (-2) = -16x\]\[-3 \times 5x = -15x\]\[-3 \times (-2) = 6\]Combine these results to get: \[40x^2 - 16x -15x + 6\]Combine like terms \(-16x\) and \(-15x\):\[40x^2 - 31x + 6\].

Step 3: Compare the Results

Expression (a) results in a linear expression, \(3x - 1\), because subtracting two linear expressions leads to another linear expression. Expression (b) results in a quadratic expression, \(40x^2 - 31x + 6\), because multiplying two linear expressions results in a quadratic expression.

Key Concepts

Distributive Property

The distributive property helps in simplifying algebraic expressions by distributing, or spreading out, a term over others. It's like sharing equally across terms inside brackets. When you see an expression like

• \((a + b)(c + d)\)

apply the distributive property by multiplying each term in the first set of parentheses with every term in the second set. This method is essential in algebra for expanding expressions, especially when dealing with binomials. In our problem, for expression (a), we used it to remove parentheses:

• \((8x - 3)-(5x-2) \Rightarrow (8x - 3 - 5x + 2)\)

In expression (b), we applied it to multiply terms, often referred to as the FOIL method: First: \(8x \times 5x\) Outer, Inner, Last: ensuring each term multiplies with each other properly. Using the distributive property makes handling algebraic expressions intuitive and more straightforward.

Like Terms

Like terms are terms that have the same variables raised to the same power. They can be easily combined to simplify an expression. Consider the expression

• \(8x - 3x + x\)

Here, all terms involving \(x\) are like terms.When simplifying, you add or subtract the coefficients, which are just the numbers in front of the variables. For example, in expression (a) from our exercise, we simplified it by combining the terms with \(x\):

• \((8x - 5x) + (-3 + 2)\)

This results in \(3x - 1\). Identifying and combining like terms can significantly reduce the complexity of algebraic expressions, making them resolvable and easier to work with.

Binomials

A binomial expression consists of two terms combined usually by a plus or minus sign. Binomials are common in algebra and often appear in parentheses as part of a larger expression, like

• \((8x-3)\)
• \((5x-2)\)

in our example. They play a crucial role in multiplication processes, notably when using the distributive property or the FOIL method to expand products of binomials. Understanding the layout and structure behind binomials aids in correctly applying operations and solving algebraic problems.Each part of a binomial has distinct roles that influence the action of how terms get distributed or combined.

Quadratic Expression

Quadratic expressions contain terms where the highest power of the variable is two. These expressions appear as results of multiplying two linear binomials, as seen in expression (b) from our sample:

• \((8x-3)(5x-2)\)

After expanding, you end up with:

• \(40x^2 - 31x + 6\)

The term \(40x^2\) indicates it’s a quadratic. The defining feature is the squared term, which influences the "U-shape," or parabola, of its graph. Quadratic expressions are pivotal in many mathematical functions and are frequently solved to find values like roots representing the equation, say when equal to zero. Recognizing how binomials transform into a quadratic expression enhances understanding and solving challenges involving second-degree polynomials.