Problem 46 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using the FOIL method, and simplify. \((4 x-5 y)(3 x-y)\)

Step-by-Step Solution

Answer

The simplified expression is \(12x^2 - 19xy + 5y^2\).

Step 1: Use FOIL Method

FOIL stands for First, Outer, Inner, Last, and is a method used to expand binomials. Apply FOIL to the expression \((4x - 5y)(3x - y)\). Begin by multiplying the First terms: \(4x \cdot 3x = 12x^2\).

Step 2: Multiply Outer Terms

Next, multiply the Outer terms: \(4x \cdot (-y) = -4xy\).

Step 3: Multiply Inner Terms

Multiply the Inner terms: \(-5y \cdot 3x = -15xy\).

Step 4: Multiply Last Terms

Now, multiply the Last terms: \(-5y \cdot (-y) = 5y^2\).

Step 5: Combine Terms and Simplify

Combine all the results from previous steps: \(12x^2 - 4xy - 15xy + 5y^2\). Next, simplify by combining like terms: \(12x^2 - 19xy + 5y^2\).

Key Concepts

Algebraic Expressions

Algebraic expressions form the cornerstone of algebra and consist of numbers, variables, and operations. These expressions can range from simple to complex, often including several terms. For example, in the exercise \((4x - 5y)(3x - y)\), each part represents a term or a combination of terms known as a polynomial.

In these expressions, variables like \(x\) and \(y\) represent unknown values and allow us to create general formulas that hold true for numerous situations. Operations include addition, subtraction, and multiplication. Understanding how to manipulate these expressions is crucial for solving algebraic equations and modeling real-world situations.

• Variables represent unknowns and can be letters such as \(x, y, z\).
• Operations combine variables and numbers in various ways.
• Expressions can be simplified by following algebraic rules.

Binomials

Binomials are specific types of algebraic expressions that consist of exactly two terms. An example from the exercise is \((4x-5y)\). Binomials are incredibly important in algebra because they often serve as building blocks for more complex problems.

Understanding binomials involves recognizing each part: the terms and the operations between them (either addition or subtraction). In solving mathematical problems, binomials are often manipulated through multiplication or division to expand or factor expressions.

• Consist of two terms separated by a plus or minus.
• Examples include expressions like \((a + b)\) or \((x - y)\).
• Crucial in operations such as polynomial multiplication and factoring.

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions that can have one or more terms, and it’s essential for expanding expressions such as binomials. The FOIL method is a common technique used for multiplying two binomials.

In the given exercise, you multiply the binomials \((4x-5y)\) and \((3x-y)\) using the FOIL method by doing the following:

• First: Multiply the first terms of each binomial, resulting in \(12x^2\).
• Outer: Multiply the outer terms, \(4x\) and \(-y\), yielding \(-4xy\).
• Inner: Multiply the inner terms, \(-5y\) and \(3x\), which results in \(-15xy\).
• Last: Finally, multiply the last terms, \(-5y\) and \(-y\), to get \(5y^2\).

After carrying out these steps, combining all the results and simplifying is crucial to get the final expression.

Like Terms

"Like terms" in algebra refer to terms that have the same variables raised to the same powers, although the coefficients may differ. Identifying and combining like terms is vital for simplifying expressions, a necessary step in solving algebra problems.

In our solution, after multiplying using the FOIL method, we get the terms: \(12x^2\), \(-4xy\), \(-15xy\), and \(5y^2\). The like terms here are \(-4xy\) and \(-15xy\), as they both contain the same variables and exponents.

• Like terms have the same variable components.
• Only the coefficients, the numerical parts, differ and are added or subtracted to simplify.
• Combining like terms helps in reducing expressions to their simplest form, such as transforming \(-4xy - 15xy\) into \(-19xy\).