Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the polynomials. \(\left(b^{2}-1\right)\left(a^{2}+2 a b+b^{2}\right)\)

Step-by-Step Solution

Answer

The product is \(b^4 + b^2a^2 + 2ab^3 - a^2 - 2ab - b^2\).

Step 1: Arrange the Polynomials

Identify the two polynomials you are multiplying: \(b^2 - 1\) and \(a^2 + 2ab + b^2\).

Step 2: Apply the Distributive Property

Use the distributive property \( (x+y)(z+w) = xz + xw + yz + yw \) to multiply \( (b^2 - 1) \) by each term in \( (a^2 + 2ab + b^2) \).

Step 3: Distribute the First Term

Multiply \(b^2\) by each term in the second polynomial: 1. \(b^2 \cdot a^2 = b^2a^2\)2. \(b^2 \cdot 2ab = 2ab^3\)3. \(b^2 \cdot b^2 = b^4\)

Step 4: Distribute the Second Term

Multiply \(-1\) by each term in the second polynomial:1. \(-1 \cdot a^2 = -a^2\)2. \(-1 \cdot 2ab = -2ab\)3. \(-1 \cdot b^2 = -b^2\)

Step 5: Combine All Terms

Combine all terms obtained from the distribution:\(b^2a^2 + 2ab^3 + b^4 - a^2 - 2ab - b^2\).

Step 6: Simplify the Expression

Reorder and combine like terms to simplify the expression:\(b^4 + b^2a^2 + 2ab^3 - a^2 - 2ab - b^2\).

Key Concepts

Distributive Property

In algebra, the distributive property is essential when dealing with expressions, particularly polynomial multiplication. It allows us to take a term or set of terms and distribute it across other terms in an expression. This property states that for any numbers or algebraic expressions,

• \( a(b + c) = ab + ac \)
• \((x+y)(z+w) = xz + xw + yz + yw \)

Applying this property, as done in the exercise, ensures that each term in the first polynomial interacts with every term in the second polynomial. This interaction expands the expression fully, simplifying the process of multiplication when working with complex polynomials.
It is like distributing or sharing the multiplication process across each term, allowing us to systematically solve polynomial multiplication problems.

Like Terms

Identifying like terms is crucial in simplifying polynomial expressions. Like terms are terms whose variables and their exponents are the same, although their coefficients can vary.

• For example, in the expression \(5x + 3x\), both terms are like terms because they share the same variable and exponent (\(x^1\)).
• However, \(3x^2\) and \(4x\) are not like terms because the exponents on \(x\) are different (\(x^2\) vs. \(x^1\)).

In polynomial multiplication exercises like the one given, once the terms have been expanded using the distributive property, identifying and combining like terms is a significant step to simplifying the result. The initial arrangement of terms might look complex, but simplifying by combining like terms makes the expression more manageable and readable.

Polynomial Expressions

Polynomial expressions consist of variables, coefficients, and exponents, organized as sums and/or differences of various terms. Each polynomial term typically involves only integers as exponents of the variables.

• A simple polynomial example is \( 3x^2 + 5x - 2 \).
• In our exercise, the polynomial \(b^2 - 1\) represents a binomial because it consists of two terms, whereas \(a^2 + 2ab + b^2\) is a trinomial, having three terms.

Understanding these structures is imperative for successfully performing algebraic operations, like multiplication and simplification, on polynomials. As polynomials grow more complex, recognizing their structure helps you break down and solve algebraic problems effectively.

Algebraic Operations

Algebraic operations are the manipulations we perform on algebraic expressions, such as addition, subtraction, multiplication, and division. In the given exercise, the main focus is on multiplication of polynomial expressions using the distributive property.

• Adding or subtracting polynomials involves combining like terms.
• Multiplying polynomials requires distributing every term in one polynomial by every term in the other polynomial.
• Dividing polynomials can be more complex and may involve factoring or using synthetic division.

Regardless of the operation, understanding the basic rules and properties of algebra is crucial. It's essential to recognize the type of operation needed so you can apply the correct method. For instance, after multiplying, simplifying by combining like terms transforms a lengthy expression into a more simplified, orderly result.