Problem 72 - College Algebra Essentials

Question

In Exercises 67–82, find each product. $$\left(7 x^{2} y+1\right)\left(2 x^{2} y-3\right)$$

Step-by-Step Solution

Answer

The product of the two binomials \((7x^{2}y+1)\) and \((2x^{2}y-3)\) is \(14x^{4}y^{2} - 19x^{2}y - 3\).

Step 1: Multiply the 'First' terms

Multiply the first terms in each binomial, which in this case are \(7x^{2}y\) and \(2x^{2}y\). The product is \(14x^{4}y^{2}\).

Step 2: Multiply the 'Outer' terms

Multiply the outer terms in each binomial. That is \(7x^{2}y\) and \(-3\). The product is \(-21x^{2}y\).

Step 3: Multiply the 'Inner' terms

Multiply the inner terms. In this case, multiply \(1\) and \(2x^{2}y\). The product is \(2x^{2}y\).

Step 4: Multiply the 'Last' terms

Multiply the last terms in both binomials. In this case, \(1\) and \(-3\). The product is \(-3\).

Step 5: Add all the products

Add all the products obtained from the previous steps to get the result of the multiplication of the two binomials: \(14x^{4}y^{2} - 21x^{2}y + 2x^2y - 3 = 14x^{4}y^{2} - 19x^{2}y - 3\).

Key Concepts

Polynomial Multiplication

Taking the first steps in algebra, one must understand polynomial multiplication. It forms the basis for many algebraic procedures.

In its essence, multiplying polynomials involves multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms. To illustrate, consider the product of two binomials \( (7x^2y + 1)(2x^2y -3) \). The first binomial has terms \(7x^2y\) and \(1\), while the second has terms \(2x^2y\) and \( -3\). We multiply every term from the first binomial by every term from the second.

It's vital to remember that when multiplying terms, we must apply the distributive law to multiply coefficients and add the exponents of like bases. This results in a new polynomial where the degree (the highest power of the variable present) is the sum of the degrees of the multiplied terms. In our example, the highest degree is \(4\), arrived at by adding the degrees of \(7x^2y\) and \(2x^2y\).

FOIL Method

The FOIL method is a specialized tool under the umbrella of polynomial multiplication, particularly tailored for binomials. FOIL stands for First, Outer, Inner, Last—referring to the order in which we multiply terms from each binomial.

For the given exercise, \( (7x^2y + 1)(2x^2y - 3) \), FOIL dictates the multiplication process as follows:

First:

Multiply the first terms of each binomial (\[ 7x^2y \times 2x^2y = 14x^4y^2 \]).

Outer:

Multiply the outer terms (\[ 7x^2y \times -3 = -21x^2y \]).

Inner:

Multiply the inner terms (\[ 1 \times 2x^2y = 2x^2y \]).

Last:

Multiply the last terms of each binomial (\[ 1 \times -3 = -3 \]).

Afterward, you combine the like terms to simplify the polynomial, leaving us with the final product \[ 14x^4y^2 - 19x^2y - 3 \]. The FOIL method is an efficient technique for binomial multiplication, providing a clear structure to avoid errors and to ensure that no terms are overlooked during multiplication.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and involve constants, variables, and arithmetic operations. They represent quantities that can vary, which we call variables.

An expression like \(7x^2y + 1\) is binomial because it contains two terms. Each term is made up of coefficients (numerical factors), variables (like \(x\) and \(y\)), and possibly exponents. Multiplying binomials, as in our example, is a practical application of algebraic expressions, transforming them into new expressions.

When teaching algebraic expressions, it's crucial to emphasize the importance of performing operations strictly according to mathematical laws such as the commutative, associative, and distributive properties. These ensure that our manipulation of these expressions remains accurate. Algebraic expressions are not just abstract concepts; they have real-world applications in various fields like engineering, physics, and economics.