Problem 53 - College Algebra with Corequisite Support

Question

A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: \((4 x+1)(8 x-3)\) where \(x\) is measured in meters. Multiply the binomials to find the area of the plot in standard form.

Step-by-Step Solution

Answer

The area of the plot in standard form is \(32x^2 - 4x - 3\).

Step 1: Identify the binomials

The given expression for the area of the plot is \((4x + 1)(8x - 3)\). It consists of two binomials, \((4x + 1)\) and \((8x - 3)\).

Step 2: Apply the distributive property

To multiply the binomials, apply the distributive property (also known as the FOIL method) to each term in the first binomial against each term in the second binomial.

Step 3: Multiply the first terms

Multiply the first terms of each binomial: \[4x \cdot 8x = 32x^2\]

Step 4: Multiply the outer terms

Multiply the outer terms of each binomial: \[4x \cdot (-3) = -12x\]

Step 5: Multiply the inner terms

Multiply the inner terms of each binomial: \[1 \cdot 8x = 8x\]

Step 6: Multiply the last terms

Multiply the last terms of each binomial: \[1 \cdot (-3) = -3\]

Step 7: Combine like terms

Add all results together and combine like terms:\[32x^2 - 12x + 8x - 3\] simplifies to \[32x^2 - 4x - 3\].

Key Concepts

Understanding Binomials

A binomial is a type of algebraic expression that contains exactly two terms. You often encounter binomials when dealing with polynomials. In our original exercise, we have two binomials:

• The first binomial is \(4x + 1\).
• The second binomial is \(8x - 3\).

Each term in a binomial can consist of a number, a variable, or both. Sometimes, the terms will have coefficients, which are the numbers that multiply the variable. In our binomials:

• The term \(4x\) has 4 as the coefficient.
• The term \(8x\) has 8 as the coefficient.
• The terms \(+1\) and \(-3\) are constant terms, meaning there are no variables present.

Working with binomials often involves multiplying them, which brings us to the next concept: the distributive property.

Applying the Distributive Property

To multiply binomials like \((4x + 1)(8x - 3)\), you use the distributive property. This property states that you need to multiply each term in one binomial by every term in the other binomial. In our example, this is often called the FOIL (First, Outer, Inner, Last) method, because it helps you keep track of all the necessary multiplications:

• **First Terms:** Multiply the first terms of each binomial: \(4x\) and \(8x\), which equals \(32x^2\).
• **Outer Terms:** Multiply the outer terms, \(4x\) and \(-3\), to get \(-12x\).
• **Inner Terms:** Multiply the inner terms, \(1\) and \(8x\), equals\(8x\).
• **Last Terms:** Finally, multiply the last terms, \(1\) and \(-3\), which gives you \(-3\).

Once calculated, all the results are combined, adding like terms to simplify the expression. Make sure each multiplied result is correctly placed according to its power, especially when dealing with variable components.

Exploring Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials to produce another polynomial. When you're multiplying binomials—which have two terms each—you typically get a quadratic polynomial. A quadratic polynomial will have a term with a degree of two, represented as \(x^2\).

Using our calculated results from the distributive property:

• Start with \(32x^2\)
• Then, combine the middle terms \(-12x\) and \(8x\), resulting in \(-4x\). When combining terms, be sure only to combine like terms (terms containing the same variable and exponent).
• Finally, add the last constant term \(-3\).

The completed polynomial \(32x^2 - 4x - 3\) is in its simplest form. This process is fundamental in algebra, helping to unify and simplify various expressions and functions.