Problem 72 - College Algebra

Question

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: Distributing the first term of the first binomial

The first term in the first binomial is \(7x^{2}y\). This term has to be multiplied with every term in the second binomial which are \(2x^{2}y\) and \(-3\). The process results in \(14x^{4}y^{2}\) and \(-21x^{2}y\)

Step 2: Distributing the second term of the first binomial

The second term in the first binomial is \(1\). This term also has to be multiplied with every term inside the second binomial: \((1)*(2x^{2}y)=2x^{2}y\) and \((1)*(-3)=-3\)

Step 3: Summing the distributed terms

Each product found in steps 2 and 3 are then added: \(14x^{4}y^{2} - 21x^{2}y + 2x^{2}y - 3\)

Step 4: Simplifying the sum of terms

Combine like terms in the result from step 3: \(14x^{4}y^{2} - 19x^{2}y - 3\)

Key Concepts

Distributive Property

The distributive property, also known as the distributive law of multiplication and addition, is a foundational concept in algebra that comes into play when we multiply a single term by a polynomial (or binomial, as in our exercise). It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then summing the products. This leads to the general formula:

\forall a, b, c \( a \times (b + c) = a \times b + a \times c \).The distributive property allows us to simplify complex algebraic expressions by expanding them into more workable parts. In our exercise, we use this property to multiply the terms of the first binomial \(7 x^{2} y+1\) with each term of the second binomial \(2 x^{2} y-3\) separately, resulting in a longer expression that's ready to be simplified further.

Combining Like Terms

After applying the distributive property, we often encounter expressions filled with multiple terms. To simplify these, we use the method of combining like terms. Like terms are terms within an algebraic expression that have the same variables raised to the same powers, although they can have different coefficients.

The process involves adding or subtracting coefficients while keeping the common variable and its exponent unchanged. For example, in the result of our polynomial multiplication, \(14x^{4}y^{2} - 21x^{2}y + 2x^{2}y - 3\), the terms \( -21x^{2}y \) and \(2x^{2}y\) are like terms and can be combined to simplify the expression. This step is crucial for reducing an expression to its simplest form, making it easier to understand and work with.

Binomial Products

Binomials are algebraic expressions containing two terms separated by a plus or minus sign, such as \(a + b\). Multiplying binomials involves using the distributive property to find the product of every term in the first binomial with every term in the second. Products of binomials often result in quadratic expressions or higher degree polynomials depending on the terms involved.

In the exercise, the product of the binomials \(7 x^{2} y+1\) and \(2 x^{2} y-3\) requires each term in the first binomial to interact with each term in the second, leading to four distinct products. These products are then combined using the distributive property to obtain the full polynomial expression before simplification.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent a particular value. In algebra, we work extensively with these expressions, learning to manipulate and simplify them using various properties and methods. In our example, \(\left(7 x^{2} y+1\right)\left(2 x^{2} y-3\right)\) is an algebraic expression that we simplify through polynomial multiplication. We use the distributive property to multiply and then combine like terms to condense the expression down to its simplest form.

Understanding how to work with algebraic expressions is crucial for solving equations, modeling real-world situations, and advancing in mathematical studies. It lays the groundwork for more complex topics in algebra and beyond.