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Chapter 5: Problem 23

Find each product. $$ (k+5)(k-5) $$

Short Answer

Expert verified
(k+5)(k-5) = k^2 - 25

Step by step solution

01

Recognize the Expression

This is a difference of squares expression of the form (a+b)(a-b). Here, 'a' is 'k' and 'b' is '5'.
02

Apply the Difference of Squares Formula

Using the formula , we have the difference of squares: (a+b)(a-b) = a^2 - b^2. For this problem, replace 'a' and 'b' with 'k' and '5' respectively.
03

Substitute and Simplify

Substitute 'k' and '5' into the formula: (k+5)(k-5) = k^2 - 5^2. Simplify further: k^2 - 25.

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Key Concepts

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

algebraic multiplication
Algebraic multiplication involves multiplying expressions that have variables. This product can get more complex when dealing with polynomial expressions or functions involving parentheses. One key strategy is understanding specific formulas or patterns, like the difference of squares, which helps you simplify the multiplication process.
polynomial expressions
A polynomial expression is formed when we have multiple terms added or subtracted together, each consisting of a variable raised to a non-negative integer power and possibly multiplied by a coefficient. For example, in the expression (k+5)(k-5), each binomial is a polynomial of degree one. Dealing with such expressions often requires specialized techniques such as the difference of squares to simplify them. When you see an expression in the form (a+b)(a-b) , you can use the formula for the difference of squares to simplify your work.

This transforms (a+b)(a-b) into a squared term minus another squared term: a^2 - b^2 .
squared terms
Squared terms are expressions where a number or variable is raised to the power of two. For instance, in our exercise, the terms k^2 and 5^2 are squared terms. Recognizing these within polynomial expressions can make your calculations more straightforward. Applying the difference of squares to (k+5)(k-5) lets us replace the polynomials with their squared forms directly: k^2 - 5^2 . Simplifying this, we get: k^2 - 25 . By identifying these patterns and using the formula for the difference of squares, you can quickly and accurately deal with complex algebraic multiplication.

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Most popular questions from this chapter

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ 3 p^{3}\left(2 p^{2}+5 p\right)\left(p^{3}+2 p+1\right) $$

Use scientific notation to calculate the answer to each problem. During the \(2007-2008\) season, Broadway shows grossed a total of \(9.38 \times 10^{8}\) dollars. Total attendance for the season was \(1.23 \times 10^{7} .\) What was the average ticket price for a Broadway show?

Multiply. $$ (2 c)\left(3 c^{2}\right) $$

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