Problem 59 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (5-w)(12+3 w) $$

Step-by-Step Solution

Answer

\(60+3w-3w^2\)

Step 1: First term multiplication

First, we multiply the first terms of the two binomials \((5*12)\) resulting in 60.

Step 2: Outer term multiplication

Next, we multiply the outer terms \((5*3w)\). This gives us \(15w\).

Step 3: Inner term multiplication

We then multiply the inner terms \(-w*12\) to get \(-12w\).

Step 4: Last term multiplication

Lastly, we multiply the last terms \(-w*3w\), leading to \(-3w^2\).

Step 5: Combining Like Terms

Combine the middle two terms, \(15w\) and \(-12w\), to get \(3w\). Our final answer then becomes \(60+3w-3w^2\)

Key Concepts

Binomials

Binomials are a fundamental part of algebra and are defined as algebraic expressions that have exactly two terms. These terms are separated by a plus sign (+) or a minus sign (-). For example, in our exercise, the expressions \(5 - w\) and \(12 + 3w\) are both binomials.
Understanding binomials is essential because they form the basis for more complex algebraic expressions and equations. When working with binomials, it's important to be comfortable with operations such as addition, subtraction, and especially multiplication.
In algebra, binomial multiplication often uses the FOIL method—which stands for First, Outer, Inner, Last. This process helps organize the multiplication of each term in the first binomial with each term in the second binomial.

Combining Like Terms

Combining like terms is a crucial concept when simplifying polynomial expressions. Like terms are terms that contain the same variable raised to the same power. In our example, \(15w\) and \(-12w\) are considered like terms because they both contain the variable \(w\) with an exponent of 1.
When combining like terms, simply add or subtract the coefficients and keep the common variable part. This step simplifies expressions, making them more manageable and easier to evaluate.

• For instance, in our solution, we combined \(15w\) and \(-12w\) to obtain \(3w\).
• This effort reduces the complexity of the expression, resulting in the more concise polynomial: \(60 + 3w - 3w^2\).

Understanding how to combine like terms is essential for both polynomial simplification and solving equations.

Polynomial Multiplication

Polynomial multiplication is the process of multiplying two polynomials together to form a new polynomial. In the case of binomials, the multiplication can be efficiently carried out using the FOIL method.
The FOIL method is specifically useful for multiplying two binomials:

• First: Multiply the first terms of each binomial. For \((5 - w)\) and \((12 + 3w)\), we multiply the terms \(5\) and \(12\) to get \(60\).
• Outer: Multiply the outer terms; thus \(5\) and \(3w\), giving \(15w\).
• Inner: Multiply the inner terms; here \(-w\) and \(12\), resulting in \(-12w\).
• Last: Multiply the last terms in each binomial; \(-w\) and \(3w\) produce \(-3w^2\).

After multiplying, the next step is to combine like terms to simplify into a single polynomial. This method ensures that each term from the first binomial is multiplied by every term in the second binomial, covering all possible products. Mastery of polynomial multiplication is essential for handling more complex algebraic expressions.