Problem 68 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Distribute First Term of First Binomial

Multiply the first term of the first binomial (which is \(x\)) by each term in the second binomial. So, \(x * 6x = 6x^2\) and \(x * 7y = 7xy\).

Step 2: Distribute Second Term of First Binomial

Multiply the second term of the first binomial (which is \(9y\)) by each term in the second binomial. So, \(9y * 6x = 54xy\) and \(9y * 7y = 63y^2\).

Step 3: Combine Like Terms

Summarise all the results from the above steps in the form \(ax^2+ bxy + cy^2\). So, the expression becomes: \(6x^2 + 61xy + 63y^2\).

Key Concepts

Binomial Expansion

When we talk about binomial expansion, we're referring to the process of multiplying two binomials. A binomial is an algebraic expression that contains two terms, like \((a + b)\) or \((x - y)\). To expand them, each term in the first binomial is multiplied by each term in the second binomial. This ensures that every possible combination of terms is created.

For example, in the exercise given, we have the binomials \((x+9y)\) and \((6x+7y)\). The first step in expanding is to individually multiply \(x\) and \(9y\) by each term in the second binomial. You'll get several results that need to be combined in the end. Binomial expansion is often visualized by the FOIL method (First, Outer, Inner, Last), but in problems like these, it's better to think of simply distributing terms.

Like Terms

In algebra, 'like terms' are terms that contain the same variables, with corresponding variables having the same exponents. Like terms can be combined to simplify expressions.

In the context of the exercise, after multiplying the binomials, you get several terms: \(6x^2\), \(7xy\), \(54xy\), and \(63y^2\). Notice that \(7xy\) and \(54xy\) are like terms because they both contain the same variables \(xy\). These terms can be combined by simply adding their coefficients, resulting in \(61xy\). Always remember that you can only combine terms that have precisely the same variables and powers.

Distributive Property

The distributive property is a key algebraic principle used in multiplication to expand expressions. It states that you can distribute a multiplication operation over an addition or subtraction operation within parentheses.

In simpler terms, if you have the expression \(a(b + c)\), it can be expanded to \(ab + ac\). This principle is what guides the entire process of binomial expansion. When you're expanding the binomial \((x+9y)(6x+7y)\), it works step-wise: \(x\) is distributed over \(6x\) and \(7y\), and similarly with \(9y\).

• The first set of multiplications involve the first term of one binomial being multiplied by all terms of the other binomial.
• The second set involves the second term of the first binomial being distributed over all terms of the second.

Each multiplication is then summed up according to the distributive property, simplifying to the final expression.