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Chapter 6: Problem 26

Simplify. $$ 4 a\left(3 a^{2}+b\right) $$

Short Answer

Expert verified
The simplified expression is \(12a^3 + 4ab\).

Step by step solution

01

Understand the Expression

The expression to simplify is \(4a(3a^2 + b)\). It involves distributing \(4a\) to each term inside the parentheses.
02

Distribute the Terms

Apply the distributive property, which states that \(a(b + c) = ab + ac\). Multiply \(4a\) by each term inside the parentheses separately: \(4a \times 3a^2\) and \(4a \times b\).
03

Simplify Each Term

For the first term: \(4a \times 3a^2 = 12a^{1+2} = 12a^3\). For the second term: \(4a \times b = 4ab\).
04

Combine the Terms

The simplified expression is the sum of the two simplified terms: \(12a^3 + 4ab\).

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Key Concepts

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

Distributive Property
When working with algebraic expressions, the distributive property is a foundational concept. It's a rule used to simplify expressions and is key in multiplying polynomials. Essentially, the distributive property allows us to remove parentheses by distributing one term across terms inside the parentheses.

In the expression given, \(4a(3a^2 + b)\), the distributive property is applied as follows:
  • Multiply \(4a\) by \(3a^2\), which gives \(4a \times 3a^2\).
  • Then multiply \(4a\) by \(b\), which results in \(4a \times b\).
This showcases the rule: \(a(b+c) = ab + ac\), which helps break down complex expressions into simpler terms.
Simplifying Expressions
Simplifying expressions is about transforming a complex equation into an easier-to-read format, without changing its value. It's a critical skill for solving algebraic problems. The goal is to combine like terms and rewrite the expression as simply as possible.

For example, consider the expression after applying the distributive property: \(12a^3 + 4ab\).
  • Each term in this expression is already simplified because there are no common factors to combine.
  • The first term, \(12a^3\), indicates a coefficient of 12 and a variable \(a\) raised to the power of 3.
  • The second term, \(4ab\), contains a coefficient of 4 and two variables, \(a\) and \(b\), in a linear relationship.
Simplifying helps in identifying the structure of the expression and in further algebraic operations.
Polynomials
Polynomials are expressions composed of variables and coefficients, combined using only addition, subtraction, and multiplication. They are a key topic in algebra.

In the given exercise, \(4a(3a^2 + b)\) is simplified to \(12a^3 + 4ab\), which is a polynomial expression.
  • The term \(12a^3\) is a monomial, having a single term with a variable raised to the power of 3.
  • The entire expression \(12a^3 + 4ab\) is a binomial, since it consists of two separate terms.
Understanding the structure of polynomials helps in recognizing how they can be manipulated and what operations can be performed for their simplification.

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Most popular questions from this chapter

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}+x^{2}-16 x-16 ; x+4 $$

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{3}+4 x^{2}+x+k\right) \div(x+2) $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{4}-6 x-8 $$

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ 2\left[p\left(x^{2}+1\right)\right]-3 r(x-1) $$

PREREQUISITE SKILL. Find the exact solutions of each equation by using the Quadratic Formula. $$ 2 x^{2}+3 x+2=0 $$
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