Question

Find the product. $$ (9 w+8)(11 w-10) $$

Short answer

The simplified product of \( (9w + 8)(11w - 10) \) is \( 99w^2 - 2w - 80 \).

Step-by-step solution

Expand the Expression

In this step, the expressions \(9w + 8\) and \(11w - 10\) should be multiplied by applying the FOIL method. \n First, multiply the first terms in each binomial: \(9w * 11w = 99w^2\). \n Then multiply the outside terms: \(9w * -10 = -90w\). \n Multiply the inside terms: \(8 * 11w = 88w\). \n Finally, multiply the last terms: \(8 * -10 = -80\). After multiplying, the expression should now be written as follows: \(99w^2 - 90w + 88w - 80\).

Simplify the Expression

Combine like terms: \n The expression now looks like this: \( 99w^2 - 2w - 80 \).

Key concepts

Multiplying Binomials with the FOIL Method

When it comes to multiplying binomials, a popular technique used is the FOIL method. This method provides a systematic approach for expanding the product of two binomials. FOIL stands for First, Outside, Inside, Last, which refers to the order in which you multiply the terms of the binomials. Let's break down these steps

• First: Multiply the first term of each binomial. In our exercise (9w + 8)(11w - 10), we multiply 9w and 11w to get 99w^2.
• Outside: Multiply the outermost terms, which are 9w from the first binomial and -10 from the second, resulting in -90w.
• Inside: Multiply the inner terms, 8 from the first binomial and 11w from the second giving us 88w.
• Last: Finally, the last terms 8 and -10 are multiplied to give -80.

For learners, it is crucial to follow this order to avoid confusion and ensure all terms are accounted for. After following the FOIL steps, we combine the results to form the expanded version of the product.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra, which involves reducing expressions into their simplest form. This process makes the expressions easier to understand and use in further calculations. The basic steps include:

• Combining like terms.
• Reducing the coefficients.
• Factoring.

In our example (9w + 8)(11w - 10), after applying the FOIL method, we get 99w^2 - 90w + 88w - 80. The next step is to combine like terms, which share the same variables and exponents. In this case, -90w and 88w are like terms and can be combined. Simplifying the expression leads to 99w^2 - 2w - 80, which is the expression in its simplest form. Understanding how to simplify expressions correctly is crucial for solving equations and inequalities.

Combining Like Terms

Combining like terms is one of the most critical steps in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. The coefficients, or numbers in front of the variable, can be different. When you combine like terms, you are essentially adding or subtracting the coefficients while keeping the variable part unchanged.
In the exercise provided, after expanding the binomials, we were left with 99w^2 - 90w + 88w - 80. The terms -90w and 88w both have the variable w raised to the first power, thus they are like terms. By combining them, we add their coefficients: -90 + 88 = -2, resulting in the combined term -2w. The final simplified expression is 99w^2 - 2w - 80. Students should practice this concept by grouping like terms together and altering their coefficients accordingly to achieve a simplified expression.