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Chapter 4: Problem 14

Simplify the algebraic expressions for the following problems. $$ 3 x^{2}(2 x+5)(3 x+1) $$

Short Answer

Expert verified
Question: Simplify the following algebraic expression: (3x^2(2x+5))(3x+1) Answer: The simplified algebraic expression is 18x^4 + 51x^3 + 15x^2.

Step by step solution

01

Distribute 3x^2 across the terms inside the parentheses

To distribute 3x^2, we need to multiply it with each term inside the parentheses (2x+5): $$ 3x^2(2x + 5) = (3x^2 \times 2x) + (3x^2 \times 5) $$
02

Perform the multiplication

Multiply the terms using exponent rules and simplification: $$ (3x^2 \times 2x) + (3x^2 \times 5) = (6x^3) + (15x^2) $$ Now we have a new expression: $$ (6x^3 + 15x^2)(3x+1) $$
03

Distribute (6x^3 + 15x^2) across the terms inside the parentheses (3x+1)

To distribute (6x^3 + 15x^2) across (3x + 1), perform the multiplication: $$ (6x^3 + 15x^2)(3x+1) = (6x^3 + 15x^2) \times 3x + (6x^3 + 15x^2) \times 1 $$
04

Perform the multiplication

Multiply the terms using exponent rules and simplification: $$ (6x^3 + 15x^2) \times 3x + (6x^3 + 15x^2) \times 1 = (18x^4) +(45x^3) + (6x^3) + (15x^2) $$
05

Combine like terms

Now we can add the like terms in the expression: $$ (18x^4) +(45x^3) + (6x^3) + (15x^2) = 18x^4 + 51x^3 + 15x^2 $$ So, the simplified algebraic expression is: $$ 18x^4 + 51x^3 + 15x^2 $$

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Key Concepts

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

Simplification
Simplification in algebra is the process of reducing expressions to a simpler form that is easier to work with. This involves performing operations such as addition, subtraction, multiplication, and division, and also reducing terms wherever possible. In the given problem, simplification was crucial from start to finish. Each step of the exercise aimed to rewrite the expression in the simplest form using operations and rules.
  • Remove unnecessary parentheses by distributing.
  • Combine like terms to simplify the expression.
  • Use arithmetic operations to reach easier expression forms.
Understanding simplification is important as it helps in solving equations faster and makes expressions more manageable.
Distributive Property
The distributive property is a key algebraic concept used to eliminate parentheses in expressions. It allows you to distribute a single term over terms inside a set of parentheses. In the problem, this property was used twice:1. Initially to distribute \(3x^2\) across \((2x + 5)\).2. Later by distributing \((6x^3 + 15x^2)\) across \((3x + 1)\).Here's how it works:
  • Multiply the term outside the parenthesis with every term inside.
  • Apply this operation to all parts of the expression systematically.
This consistent application enables us to rewrite the expression without any parentheses and prepares it for further simplification by gathering like terms.
Exponent Rules
Exponent rules are essential when dealing with algebraic expressions, especially involving powers like in this problem. These rules simplify the multiplication and division of terms with exponents:
  • Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power Rule: \((a^m)^n = a^{m\cdot n}\)
  • Power of a Product Rule: \((ab)^n = a^n \times b^n\)
In the solution presented, the product of powers rule was pivotal. When we multiplied terms like \(3x^2\) by \(2x\), we added the exponents of \(x\), transforming it into \(6x^3\). Proper application of these rules simplifies complex expressions and is a foundational tool in algebra.

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Most popular questions from this chapter

For the following problems, simplify each of the algebraic expressions. $$ 5[4(6 x-3)+x]-2 x-25 x+4 $$

For the following problems, perform the multiplications and combine any like terms. $$ a b^{2}\left(a^{2}-2 b\right)\left(a+b^{4}\right) $$

For the following problems, simplify each of the algebraic expressions. $$ -8(3 a+2) $$

For the following problems, answer the question of how many. $$ (a+b) \text { 's in } 12(a+b) ? $$

Find the product. \((x-8)^{2}\).
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