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Chapter 6: Problem 193

Multiply. $$3\left(p^{2}+10 p+25\right)$$

Short Answer

Expert verified
3p^2 + 30p + 75

Step by step solution

01

Distribute the Constant

Distribute the constant 3 to each term inside the parentheses. This means you will multiply 3 by each term in the polynomial \(p^2 + 10p + 25\).
02

Multiply the Terms

First, multiply 3 by \(p^2\): \(3 \times p^2 = 3p^2\). Next, multiply 3 by 10p: \(3 \times 10p = 30p\). Finally, multiply 3 by 25: \(3 \times 25 = 75\).
03

Combine the Results

Combine all the terms you have multiplied: \(3p^2 + 30p + 75\). This is the final answer.

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Key Concepts

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

The Distributive Property
The distributive property is a key concept in algebra. It allows you to simplify expressions by distributing a constant or a variable across terms inside parentheses. For example, with the expression 3(p^2 + 10p + 25), you distribute the 3 to each term inside the parentheses.
To do this, multiply 3 by each of the terms separately:
  • 3 times the first term, ,
  • 3 times the second term, 10p,
  • and 3 times the third term, 25.
This method breaks down complex expressions into more manageable parts, making it easier to solve problems.
In our example, applying the distributive property simplifies the expression to 3p^2 + 30p + 75.
The key benefit of the distributive property is that it helps to simplify calculations and combine like terms, aiding in more straightforward problem-solving.
Monomials
Monomials are algebraic expressions consisting of a single term. They can include constants, variables, or a combination thereof, but they do not involve addition or subtraction.
Examples of monomials include:
  • 5 (a simple constant)
  • p (a single variable)
  • 3p^2 (a constant multiplied by a variable raised to a power)
In the given exercise, we operate on the monomial '3' and distribute it across the polynomial inside the parentheses. Each multiplication step generates new monomials:
  • 3 times p^2 gives 3p^2,
  • 3 times 10p gives 30p,
  • and 3 times 25 gives 75.
Understanding monomials is crucial because they form the building blocks of more complex algebraic expressions. Recognizing and working with them is an essential skill in mastering algebra.
Algebraic Expressions
Algebraic expressions consist of terms that include variables, constants, and coefficients. These expressions can be simple or complex, including addition, subtraction, multiplication, and division.
In the exercise example, the expression inside the parentheses is an algebraic expression: .
This expression includes:
  • p^2, a term where a variable is squared,
  • 10p, a term with a variable and a coefficient,
  • and 25, a constant term.
By multiplying each term in this algebraic expression by 3, we simplify the overall expression.
Algebraic expressions are fundamental in mathematics because they represent real-world situations, allowing us to formulate and solve equations.
The process of simplifying these expressions, often using properties like the distributive property, is a significant part of working with algebra.

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\frac{52 p^{5} q^{4}+36 p^{4} q^{3}-64 p^{3} q^{2}}{4 p^{2} q}$$

Divide each polynomial by the binomial. $$\left(y^{2}+7 y+12\right) \div(y+3)$$

Divide each polynomial by the monomial. $$\frac{45 y+36}{9}$$

Divide each polynomial by the monomial. $$\left(35 x^{4}-21 x\right) \div(-7 x)$$

Divide each polynomial by the binomial. $$\left(x^{2}+2 x-30\right) \div(x-5)$$
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