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Chapter 5: Problem 104

Multiply: $$(x+a)(x-b)(x-a)(x+b)$$

Short Answer

Expert verified
The simplified result is \(x^4 - a^2 b^2\).

Step by step solution

01

- Group and Arrange

Group the expression in pairs to simplify the multiplication process: \((x+a)(x-a)\) and \((x-b)(x+b)\).
02

- Use the Difference of Squares Formula

Apply the difference of squares formula \( (x+y)(x-y) = x^2 - y^2 \) to each pair. For the first pair \((x+a)(x-a)\), we get: \((x+a)(x-a) = x^2 - a^2\) Similarly, for \((x-b)(x+b)\), we get: \((x-b)(x+b) = x^2 - b^2\).
03

- Combine Simplified Expressions

With the results from Step 2, we now have: \((x^2 - a^2)(x^2 - b^2)\).
04

- Reapply the Difference of Squares

Apply the formula again: \[ (X+Y)(X-Y) = (X^2 - Y^2) \] Where \(X = x^2\) and \(Y = a^2\), \(Z = b^2\). Then: \[ (x^2 - a^2)(x^2 - b^2) = x^4 - (a^2)(b^2) \]

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Key Concepts

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

Polynomial Multiplication
Polynomial multiplication is a fundamental concept in algebra, involving the product of two or more polynomials. In our exercise, we are multiplying four expressions:
  • (x+a)
  • (x-b)
  • (x-a)
  • (x+b)
To simplify this multiplication, we can group them into pairs:
  • (x+a)(x-a)
  • (x-b)(x+b)
By using the difference of squares formula, each pair can be simplified before being multiplied together. This makes the problem much easier to handle.
Throughout the process, it's important to keep track of each term and ensure that we fully simplify at each step.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. The given problem is an example of handling and simplifying such expressions. Let's break down the steps to simplify our algebraic expression.First, we recognize that we can group the terms to make our work easier:
  • (x+a)(x-a)
  • (x-b)(x+b)
Next, we use the difference of squares formula:
  • (x+a)(x-a) becomes x^2 - a^2
  • (x-b)(x+b) becomes x^2 - b^2
At this stage, our expression is now (x^2 - a^2)(x^2 - b^2). Simplifying further, we reapply the difference of squares formula to get x^4 - a^2b^2.
Handling algebraic expressions effectively requires recognizing patterns and applying formulas.
Algebraic Formulas
Algebraic formulas are used to simplify and solve expressions and equations. In the original problem, the formula we repeatedly use is the difference of squares formula: (X+Y)(X-Y) = X^2 - Y^2. This formula states that the product of a sum and a difference of the same terms equals their difference squared.
Let's apply this formula step by step in our exercise:1. Group the expressions:
  • (x+a)(x-a) becomes x^2 - a^2
  • (x-b)(x+b) becomes x^2 - b^2
2. Combine the results:(x^2 - a^2)(x^2 - b^2)3. Reapply the formula:Let X = x^2 and Y = a^2, Z = b^2, then the product looks like:(X+Y)(X-Z) = X^2 - (YZ)So, we get:(x^4 - a^2b^2)
Understanding and mastering algebraic formulas like this is key to solving polynomial multiplication

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