Problem 53 - Beginning and Intermediate Algebra

Question

Use FOIL to multiply. $$(6-5 y)(2-y)$$

Step-by-Step Solution

Answer

Using FOIL to multiply \((6-5y)(2-y)\), we get the final expression: \(5y^2 - 16y + 12\).

Step 1: Write down the given expression

The given expression is: \[(6-5y)(2-y)\]

Step 2: Apply FOIL method

Following the FOIL method, we have: - First terms: \(6 \times 2\) - Outer terms: \(6 \times (-y)\) - Inner terms: \((-5y) \times 2\) - Last terms: \((-5y) \times (-y)\)

Step 3: Perform the multiplications

Now, let's perform the multiplications: - First terms: \(6 \times 2 = 12\) - Outer terms: \(6 \times (-y) = -6y\) - Inner terms: \((-5y) \times 2 = -10y\) - Last terms: \((-5y) \times (-y) = 5y^2\)

Step 4: Combine the terms

Now, let's combine all the terms: \(12 - 6y - 10y + 5y^2\) Now, let's combine the like terms \(-6y\) and \(-10y\): \(12 - 16y + 5y^2\)

Step 5: Write the final expression

After combining the terms, the final expression is: \[\boxed{5y^2 - 16y + 12}\]

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomial expressions together. In algebra, a polynomial is an expression that can include constants, variables, and exponents, all combined using addition, subtraction, and multiplication.
When multiplying polynomials, each term in the first polynomial must be multiplied by each term in the second polynomial. This often results in a new polynomial with more terms.

The FOIL method is commonly used for multiplying two binomials, which are polynomials with two terms. FOIL stands for First, Outer, Inner, Last, indicating the order in which you multiply the terms:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of each binomial.
• Inner: Multiply the inner terms of each binomial.
• Last: Multiply the last terms of each binomial.

Once all the products are found, they are combined to form a single expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (letters that represent numbers), and operations (like addition and multiplication). An algebraic expression doesn't have an equals sign, so you aren't solving for a value but manipulating the expression itself.

In the expression \((6 - 5y)(2 - y)\), each component is an algebraic expression. When these expressions are multiplied, the result is another expression that needs simplification.
Algebraic expressions can be simplified by removing parentheses and combining like terms. This results in a cleaner and often simpler expression.

Understanding how to manipulate algebraic expressions is fundamental in algebra. It helps solve equations and simplify expressions, making them easier to work with.By mastering algebraic expressions, students can perform operations more efficiently and understand deeper concepts in mathematics.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions by merging terms that have the same variables and exponents. This process reduces the number of terms in an expression, making it easier to read and solve.
For instance, in the expression \(12 - 6y - 10y + 5y^2\), the terms \(-6y\) and \(-10y\) can be combined because they both have the variable \(y\) with the same exponent.

Combining like terms involves:

• Identifying terms that have exactly the same variable parts.
• Adding or subtracting their coefficients—the numbers in front of the variables.

In our example, you combine \(-6y\) and \(-10y\) to get \(-16y\).
This simplification step is essential for achieving the final, clean expression \(5y^2 - 16y + 12\), which helps in solving algebra problems more effectively.