Problem 52 - Algebra A Combined Function

Question

Multiply. \(\left(x^{2}-4\right)^{2}\)

Step-by-Step Solution

Answer

\( (x^2 - 4)^2 = x^4 - 8x^2 + 16 \).

Step 1: Identify the Binomial

The given expression is \((x^2 - 4)^2\). Here, \(x^2 - 4\) is the binomial that we need to square.

Step 2: Use the Binomial Square Formula

Recall that the square of a binomial \( (a - b)^2 \) is given by the formula \( a^2 - 2ab + b^2 \). In this case, identify \(a = x^2\) and \(b = 4\).

Step 3: Apply the Formula

Using the formula \((a-b)^2 = a^2 - 2ab + b^2\), substitute \(a = x^2\) and \(b = 4\) into the formula. Calculate each term: \(a^2 = (x^2)^2 = x^4\), \(2ab = 2 \cdot x^2 \cdot 4 = 8x^2\), and \(b^2 = 4^2 = 16\).

Step 4: Combine the Terms

Combine the terms obtained: \(x^4 - 8x^2 + 16\). This is the expanded form of \((x^2 - 4)^2\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves finding the product of two polynomial expressions. At its core, it's about distributing each term in one polynomial to every term in the other polynomial. In this exercise, although it appears we are dealing with just a single polynomial, the process of squaring ((x^2 - 4)^2) implicitly involves multiplication. Here, we treat ((x^2 - 4)^2) as ((x^2 - 4) imes (x^2 - 4)). When expanded, each term of the first binomial is multiplied by each term of the second binomial. This is similar to how we multiply numbers but involves variables and constants. Steps to Multiply Polynomials: - Use the distributive property to expand each term. - Multiply coefficients (numbers) and add exponents for like bases. - Combine like terms to simplify the expression. In the exercise, the use of the binomial formula streamlines this multiplication, but effectively it is the same as performing the direct expansion using these multiplication steps.

Binomial Expansion

Binomial expansion refers to expressing the power of a binomial as a sum of terms in simpler form using a well-known identity. The formula ((a - b)^2 = a^2 - 2ab + b^2) used in the solution is a part of the binomial expansion for squaring binomials. This formula not only simplifies the multiplication but also gives a direct result quickly. Here are the main steps to understand binomial expansion: - Recognize the binomial: In our case, it’s ((x^2 - 4)). - Identify a and b: Let (a = x^2) and (b = 4). - Apply the binomial square formula: Calculate each component (a^2, 2ab,) and (b^2). - Write the expanded form by combining all calculated components together. Using the binomial formula ensures that we don't miss any terms and follow an organized approach to arrive at our answer: (x^4 - 8x^2 + 16).

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (such as plus and minus) that represents a mathematical object. In the context of this exercise, ((x^2 - 4)) is an algebraic expression, particularly a binomial. Key Components of Algebraic Expressions:

• Variables: Symbols like x, y, or z that represent numbers.
• Coefficients: Numbers that multiply the variable, such as 4 in (4x).
• Constants: Fixed numbers, such as 4 in the binomial (x^2 - 4).

Working with algebraic expressions often involves simplifying or transforming them following certain mathematical rules. In this exercise, simplifying ((x^2 - 4)^2) led us to the expanded form (x^4 - 8x^2 + 16). Here, knowing how to identify and manipulate each part of the expression is crucial for performing algebraic operations accurately.