Problem 68 - College Algebra Essentials

Question

In Exercises 67–82, find each product. $$(x+9 y)(6 x+7 y)$$

Step-by-Step Solution

Answer

The product of \((x+9y)(6x+7y)\) is \(6x^2 + 61xy + 63y^2\).

Step 1: Apply the Distributive Property

We begin by applying the distributive property, also known as the FOIL method. FOIL stands for First, Outer, Inner, Last.

Step 2: Multiply the First Terms

First, multiply the first terms from each binomial: \(x * 6x = 6x^2\).

Step 3: Multiply the Outer Terms

Next, we multiply the terms on the outside, or the first term of the first binomial and the second term of the second: \(x * 7y = 7xy\).

Step 4: Multiply the Inner Terms

Then, we multiply the inner terms, or the second term of the first binomial and the first term of the second: \(9y * 6x = 54xy\).

Step 5: Multiply the Last Terms

Finally, multiply the last terms from each binomial: \(9y * 7y = 63y^2\).

Step 6: Combine Like Terms

We then add together the results of each multiplication: \(6x^2 + 7xy + 54xy + 63y^2\). The like terms here are \(7xy\) and \(54xy\). Combining these gives: \(6x^2 + 61xy + 63y^2\).

Key Concepts

Distributive Property

The distributive property is essential in simplifying and solving algebraic expressions, including polynomials. It is a rule that allows us to multiply a single term by each term within a parenthesis. In mathematical terms, for any numbers or variables, it is expressed as:

• \( a(b + c) = ab + ac \)

This property is fundamental when working with binomials, like in our exercise, because it helps in breaking down complex multiplication into smaller, more manageable parts.

In the given exercise, we employed the distributive property when expanding the product \((x + 9y)(6x + 7y)\). This means we distribute each term in the first binomial over both terms in the second binomial. This step-by-step expansion is often called the FOIL method, specifically used for binomials.

FOIL Method

The FOIL method is a specific application of the distributive property when multiplying two binomials. FOIL stands for First, Outer, Inner, and Last, describing the sequence in which you multiply the terms:

• First: Multiply the first terms in each binomial, as in \( x \times 6x \).
• Outer: Multiply the outermost terms, like \( x \times 7y \).
• Inner: Multiply the inner terms, or the second of the first binomial and the first of the second, such as \( 9y \times 6x \).
• Last: Multiply the last terms in each binomial, for example, \( 9y \times 7y \).

Using the FOIL method helps ensure that all pairs of terms from each binomial are multiplied together. This method is particularly useful as it provides a structured and detailed way to approach polynomial multiplication, ultimately simplifying the multiplication process and reducing mistakes.

Combining Like Terms

After expanding and multiplying terms, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable parts raised to the same powers. These terms can be added or subtracted just like numbers.

In our exercise, after completing the FOIL process, we had terms such as \( 6x^2 \), \( 7xy \), \( 54xy \), and \( 63y^2 \). Here, \( 7xy \) and \( 54xy \) are like terms because both have the variable part \( xy \).

Combining these gives us \( 61xy \), while the terms \( 6x^2 \) and \( 63y^2 \) remain independent because they don't have like terms to combine with. Simplifying in this way is crucial for expressing the polynomial in a more concise and understandable form.

Polynomial Expressions

Polynomial expressions are algebraic expressions consisting of sums and differences of monomials, where each monomial is a product of a number (coefficient) and the variable(s) raised to a non-negative integer power. In simpler terms, a polynomial can have terms like \( ax^2 \), \( bx \), and \( c \), where each term consists of a coefficient and one or more variables.

• A polynomial of degree 1 is a linear polynomial.
• Degree 2 is a quadratic polynomial.
• Degree 3 is a cubic polynomial, and so on.

The expression \( 6x^2 + 61xy + 63y^2 \) from our exercise is a polynomial with three terms, each different in their respective variables and coefficients. Learning to effectively manipulate polynomial expressions, like through multiplication or simplification, is a foundational skill for deeper algebraic studies, calculus, and many real-world applications.