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Chapter 10: Problem 36

Multiply the polynomials. $$(y+5)(y-5)$$

Short Answer

Expert verified
(y+5)(y-5) simplifies to y^2 - 25.

Step by step solution

01

- Recognize the Structure

Identify that the expression (y+5)(y-5) is a difference of squares. The difference of squares formula is (a+b)(a-b) = a^2 - b^2.
02

- Assign Variables

Assign the variables in the formula: Let a = y and b = 5.
03

- Apply the Difference of Squares Formula

Substitute the assigned variables into the formula: (y+5)(y-5) becomes y^2 - 5^2.
04

- Simplify the Expression

Simplify the squared term: y^2 - 5^2 becomes y^2 - 25.

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Key Concepts

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

Polynomial Multiplication
The process of polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial.
This often requires the distributive property. In the given exercise, we are multiplying \(y+5\) by \(y-5\).
As we will see, the expression represents a special pattern known as the difference of squares.
When dealing with polynomial multiplication, you need to:
  • Identify each polynomial, and review their terms.
  • Apply the distributive property, meaning each term in the first polynomial must multiply each term in the second polynomial.
  • Combine like terms where possible.
Remember, this process often reveals recognizable patterns, which simplifies calculations.
Algebraic Identities
Algebraic identities are key formulas and patterns that are always true for any values of the variables involved.
These identities simplify complex algebraic expressions.
For the given exercise, we used the difference of squares identity: \(a^2 - b^2 = (a+b)(a-b)\)
To apply this identity, follow these steps:
  • Identify the form of the expression: \(a+b\) and \(a-b\).
  • Assign appropriate variables to the terms.
  • Substitute these variables in the identity to simplify the expression to \(a^2 - b^2\).
The difference of squares pattern is useful because it turns products into simpler subtractions, which are easier to handle in algebra.
Simplifying Expressions
Simplifying expressions means rewriting them in their simplest and most convenient form.
This often involves combining like terms, applying algebraic identities, and performing arithmetic operations.
In our example:
  • We started with \( (y+5)(y-5) \).
  • Recognized it as a difference of squares.
  • Applied the difference of squares formula, substituting the variables: \(y^2 - 5^2\).
  • Simplified further to get: \(y^2 - 25\).
Breaking down expressions step-by-step helps you see the simple components of complex algebraic forms. This process makes solving and understanding algebra much easier.

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Most popular questions from this chapter

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