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Chapter 5: Problem 54

Multiply. $$ \left(3-c^{2} d^{2}\right)\left(4+c^{2} d^{2}\right) $$

Short Answer

Expert verified
The simplified expression is \(9 - c^4 d^4\).

Step by step solution

01

- Recognize the Pattern

Notice that the expression is of the form \((a - b)(a + b)\), which is a difference of squares. Here, \(a = 3\) and \(b = c^2 d^2\). The difference of squares formula is given by \(a^2 - b^2\).
02

- Apply the Difference of Squares Formula

Substitute the identified values into the formula: \(a^2 - b^2 = 3^2 - (c^2 d^2)^2\).
03

- Simplify the Exponents

Calculate the squares: \(3^2 = 9\) and \((c^2 d^2)^2 = c^4 d^4\). This simplifies the expression to \(9 - c^4 d^4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Multiplication
Polynomial multiplication involves multiplying two polynomials to obtain a single polynomial. When multiplying polynomials, you distribute each term in the first polynomial by each term in the second polynomial.
In our problem, we begin with the polynomials \((3 - c^2 d^2)\) and \(4 + c^2 d^2)\). But instead of distributing each term, we recognize a specific pattern.
Difference of Squares Formula
The difference of squares is a special multiplication pattern for polynomials. It states: \((a - b)(a + b) = a^2 - b^2\).
In this format, \((3 - c^2 d^2)(4 + c^2 d^2)\), we identify \(a = 3\) and \(b = c^2 d^2\).
Applying the difference of squares formula, we get \(a^2 - b^2\).
Plugging in the values: \((3)^2 - (c^2 d^2)^2\).
Simplifying Exponents
Once we've substituted the values into the difference of squares formula, we need to simplify the exponents.
First, calculate \(a^2\). Here, \(a = 3\), so \(3^2 = 9\).
Next, for \((c^2 d^2)^2\), use the power of a power rule: \( (x^m)^n = x^{m \times n}\). Thus, \( (c^2 d^2)^2 = c^{2 \times 2} d^{2 \times 2} = c^4 d^4\).
Putting it all together, the expression simplifies to: \(9 - c^4 d^4\).

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