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Chapter 4: Problem 84

Find each product. $$ [(m+p)-5][(m+p)+5] $$

Short Answer

Expert verified
(m^2 + 2mp + p^2 - 25)

Step by step solution

01

Recognize the Formula

Recognize that the given expression has the form of a difference of squares, which is given by [a鈭抌][a+b]=a^2鈭抌^2. Here, a = m+p and b=5.
02

Substitute the Values

Substitute the values of a and b into the difference of squares formula: (m+p)^2 - 5^2.
03

Calculate Each Term

Calculate (m+p)^2 and 5^2 . (m+p)^2 = m^2 + 2mp + p^2. 5^2 = 25.
04

Combine the Terms

Combine the terms obtained in step 3 to get the final expression: m^2 + 2mp + p^2 - 25.

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Key Concepts

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

Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. It's a way to represent numbers and operations concisely. For example, in the exercise given [(m+p)-5][(m+p)+5], the expression consists of addition, subtraction, and variables m and p. Understand these key points for algebraic expressions:
  • Variables: Symbols like m and p that represent numbers.

  • Constants: Fixed numbers like 5.

  • Operators: Actions like addition (+) and subtraction (-).
Knowing how to work with these components is essential for solving algebra problems. With practice, you'll get better at identifying and manipulating these parts of an expression.
Factoring
Factoring is a process of breaking down an algebraic expression into simpler components that can be multiplied together to get the original expression. Here鈥檚 why factoring is useful:
  • Simplifies expressions and equations.

  • Makes it easier to solve equations.
In the provided exercise, [(m+p)-5][(m+p)+5], we use the difference of squares formula, which is a special type of factoring. Recognize the formula: \[ \text{{difference of squares: }} (a-b)(a+b) = a^2 - b^2 \] By understanding this, you'll see that \[ [(m+p)-5][(m+p)+5] \] can be written as \[ (m+p)^2 - 5^2 \] This step simplifies the problem and makes it easier to calculate.
Polynomials
Polynomials are expressions that consist of variables and coefficients, entailing only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The final form of our exercise solution is a polynomial: \[ m^2 + 2mp + p^2 - 25 \] Let鈥檚 break down what a polynomial consists of:
  • Terms: Parts of the polynomial separated by + or - signs.

  • Coefficients: Numbers before the variables, such as 2 in 2mp.

  • Exponents: The power to which a variable is raised, like the 2 in m^2.
Understanding polynomials is crucial because they appear frequently in algebra. Recognize and manipulate polynomials to solve and simplify problems effectively.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{-3 x^{4} y^{6}}{15 x^{-6} y^{7}}\right)^{-3} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 m^{2} p^{3}\right)^{2}\left(4 m^{2} p\right)^{-2}}{\left(-3 m p^{4}\right)^{-1}\left(2 m^{3} p^{4}\right)^{3}} $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ \left(4 x-\frac{2}{3}\right)\left(4 x+\frac{2}{3}\right) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{5 m^{4} n^{-3}}{m^{-5} n^{2}}\right)^{-2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(4^{-4}\right)^{-2} $$
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