Problem 69 - College Algebra

Question

Find each product. $$ (x-3 y)(2 x+7 y) $$

Step-by-Step Solution

Answer

The product of the two binomials \(x-3y)\) and \(2x+7y)\) is \(2x^2 + xy - 21y^2\).

Step 1: Calculating the 'First' in FOIL

Multiply the first terms in each binomial: \(x\) and \(2x\) which gives \(2x^2\).

Step 2: Calculating the 'Outer' in FOIL

Multiply the outer terms: \(x\) from the first binomial and \(7y\) from the second binomial giving \(7xy\)

Step 3: Calculating the 'Inner' in FOIL

Multiply the inner terms: \(-3y\) from the first binomial and \(2x\) from the second. This results in \(-6xy\).

Step 4: Calculating the 'Last' in FOIL

Multiply the last terms in each binomial. This involves \(-3y\) and \(7y\) which gives \(-21y^2\).

Step 5: Simplifying the Result

Combine the results from the previous steps. The result is \(2x^2 + 7xy - 6xy -21y^2\). Simplify this to get \(2x^2 + xy - 21y^2\).

Key Concepts

Binomial Multiplication

When we talk about binomial multiplication, we are referring to the process of multiplying two binomials together. A binomial is an algebraic expression that has two terms, such as \(x - 3y\) or \(2x + 7y\), which are separated by a plus or minus sign.

One of the most straightforward methods to multiply binomials is the FOIL method, which stands for First, Outer, Inner, Last. This refers to the order in which we multiply the terms of the binomials:

• First: Multiply the first terms in each of the binomials.
• Outer: Multiply the outermost terms of the binomial.
• Inner: Multiply the innermost terms.
• Last: Multiply the last terms in each binomial.

Once we carry out these steps, we combine like terms to simplify the resulting expression. Each multiplication and combination of terms is crucial in achieving the final simplified product.

Polynomial Multiplication

Polynomial multiplication expands upon the principles of binomial multiplication. Polynomials are algebraic expressions that can contain more than two terms. To multiply polynomials, we use the distributive property to ensure each term in the first polynomial is multiplied by each term in the second polynomial.

For instance, if we were to multiply a trinomial by a binomial, we would still apply the FOIL method and then continue distributing the remaining terms accordingly. The process involves several multiplication steps, followed by combining like terms, similar to binomial multiplication. In this methodical approach, attention to detail is crucial to prevent errors during multiplication or combining terms.

Remember that the number of terms in the final polynomial will be contingent on the distinct terms created during the multiplication process before simplification.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra, which allows us to express complex equations in a more understandable and concise form. This involves combining like terms and reducing expressions to their simplest form.

Like terms are terms that have the same variables raised to the same powers, though they may have different coefficients. In simplifying, we add or subtract the coefficients of like terms. For example, \(2x^2 + 7xy - 6xy - 21y^2\) simplifies to \(2x^2 + xy - 21y^2\), where \(7xy - 6xy\) are like terms and were combined.

Always look out for opportunities to combine terms and reduce coefficients, and remember to follow arithmetic rules and the order of operations. Fully simplified expressions are not only easier to understand, but they are also essential when solving for variables and working with complex algebraic operations.