Problem 70 - College Algebra

Question

Find each product. $$ (3 x-y)(2 x+5 y) $$

Step-by-Step Solution

Answer

\$6 x^2 + 13 x y - 5 y^2$

Step 1: First (F) in FOIL

Find product of the first terms in both binomials. This is done by multiplying the first element of the first binomial \(3 x\) by the first element of the second binomial \(2 x\), which yields \(6 x^2\).

Step 2: Outside (O) in FOIL

Find product of the outside terms, which are the first term of the first binomial and the last term of the second binomial. Multiply \(3 x\) and \(5 y\) to get \(15 x y\).

Step 3: Inside (I) in FOIL

Find product of the inside terms, which are the last term of the first binomial and the first term of the second binomial. Multiply \(-y\) and \(2 x\) to get \(-2 x y\).

Step 4: Last (L) in FOIL

Find product of the last terms in both binomials. Multiply \(-y\) and \(5 y\) to get \(-5 y^2\).

Step 5: Combining Like Terms

Combine all from step 1 to step 4 results together. Substituting all values, we get: \(6 x^2+ 15 x y - 2 x y - 5 y^2\).\nThen, combine the like terms \(15 x y\) and \(-2 x y\) to yield: \(6 x^2 + 13 x y - 5 y^2\).

Key Concepts

Binomial Multiplication

When we talk about binomial multiplication, we're discussing how to multiply two binomials to find a single expression. Binomials are algebraic expressions that contain two terms. For example, in the exercise given, the two-binomial expressions are \((3x - y)\) and \((2x + 5y)\).

A common method to multiply these binomials is the FOIL Method. FOIL stands for First, Outside, Inside, and Last, which refers to the order in which you multiply the terms of the binomials.

• **First**: Multiply the first terms of each binomial, which, in this case, are \(3x\) and \(2x\). This gives us \(6x^2\).
• **Outside**: Multiply the outer terms \(3x\) and \(5y\), resulting in \(15xy\).
• **Inside**: Multiply the inner terms, \(-y\) and \(2x\), which gives \(-2xy\).
• **Last**: Multiply the last terms of each binomial: \(-y\) and \(5y\) to give \(-5y^2\).

Each product from these steps is then combined to form a single expression. Understanding this process is crucial for simplifying more complex polynomial expressions.

Polynomials

Polynomials are expressions that consist of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents. They are versatile in mathematics as they can represent a wide range of functions and equations.

In our exercise, the end goal of binomial multiplication is to construct a polynomial. After applying the FOIL method to multiply \((3x - y)\) and \((2x + 5y)\), the resulting polynomial is \(6x^2 + 15xy - 2xy - 5y^2\).

Polynomials can be as simple as a single term or involve many terms of various degrees. The degree of a polynomial is the highest power of the variable within the expression. In our case, the highest degree is 2, coming from the \(6x^2\) term, making it a quadratic polynomial. Recognizing the structure and degree of a polynomial helps immensely in algebraic manipulation and problem-solving.

Combining Like Terms

Once you have derived the expression from your multiplication process, the next step is to simplify it by combining like terms. This step is essential to achieving the simplest and most readable form of your expression.

Like terms refer to those terms in a polynomial that have the same variable raised to the same power. In our polynomial \(6x^2 + 15xy - 2xy - 5y^2\), the like terms are \(15xy\) and \(-2xy\).

• To combine these, you simply add or subtract the coefficients of these terms. Thus, \(15xy - 2xy = 13xy\).

The resulting polynomial, after combining like terms, is \(6x^2 + 13xy - 5y^2\), which is the simplest form of the expression. By practicing this skill, students can more efficiently simplify complex expressions and better understand algebraic operations.