Problem 53 - College Algebra

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 is \(32x^2 - 4x - 3\).

Step 1: Distribute the First Term of the First Binomial

To multiply the binomials \((4x + 1)(8x - 3)\), start by distributing the first term \(4x\) of the first binomial to each term in the second binomial.\[ 4x \times 8x = 32x^2 \] and \[ 4x \times (-3) = -12x \].

Step 2: Distribute the Second Term of the First Binomial

Now distribute the second term \(+1\) of the first binomial to each term in the second binomial.\[ 1 \times 8x = 8x \] and \[ 1 \times (-3) = -3 \].

Step 3: Combine Like Terms

Now, add all these terms together: \[ 32x^2 - 12x + 8x - 3\]. Combine the like terms, \(-12x\) and \(+8x\), to simplify the expression.\[ 32x^2 - 4x - 3 \].

Step 4: Final Result

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

Key Concepts

Binomial Distributive Property

The binomial distributive property is a fundamental concept in algebra that helps us extend the multiplication of two binomial expressions. When we have two binomials, such as

• \((a + b)(c + d)\)

we use this property to multiply each term in the first binomial by each term in the second binomial. This ensures that every possible combination of terms is accounted for. For example, with

• \((4x + 1)(8x - 3)\)

the first term \(4x\) is distributed across the second binomial:

• \[4x \times 8x = 32x^2\]
• \[4x \times (-3) = -12x\]

Next, the second term \(+1\) is also distributed:

• \[1 \times 8x = 8x\]
• \[1 \times (-3) = -3\]

Using the distributive property properly is key to ensuring that you don’t miss any terms in the final polynomial.

Combining Like Terms

After distributing the terms, the next crucial step is combining like terms. Like terms are terms that have the same variable raised to the same power. In the expression resulting from our distribution

• \[32x^2 - 12x + 8x - 3\]

we can see that \(-12x\) and \(+8x\) are like terms because they both contain the variable \(x\). To combine them, simply add or subtract their coefficients:

• \[-12x + 8x = -4x\]

This gives us a new polynomial:

• \[32x^2 - 4x - 3\]

Combining like terms is a way of simplifying polynomial expressions, making them easier to understand and work with. Always remember to maintain the right signs while combining terms.

Standard Form of a Polynomial

A polynomial is in standard form when it is ordered from highest to lowest degree based on the exponents of its variable parts. This means you'll arrange the terms so that the powers of \(x\) (or any variable) go from largest to smallest. For example,

• \[32x^2 - 4x - 3\]

is already in standard form because the term with \(x^2\) comes first, followed by the \(x\) term, and finally the constant term. Having polynomials in standard form makes it easier to read, compare, and apply further operations such as adding, subtracting, or factoring. It's a good practice to always simplify and arrange your polynomial expressions in this way. Doing so can also help reduce errors in lengthy algebraic operations.