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Chapter 0: Problem 41

simplify each expression by factoring. $$ 3 x(x+1)^{3 / 2}-6(x+1)^{1 / 2} $$

Short Answer

Expert verified
The simplified form of the expression is \((x+1)^{1/2} * 3(x - 1)(x + 2)\)

Step by step solution

01

Identify common factors

Firstly, you should notice that both terms in the expression have \((x+1)^{1/2}\) as a common factor, where the first term has this factor squared. So, its power is higher.
02

Factor out the common factors

Now, you can factor out this common factor by dividing each term by it: \[3x(x+1)^{3 / 2}-6(x+1)^{1 / 2} = (x+1)^{1/2} (3x(x+1)-6)\]
03

Simplify the factored expression

Next, you should simplify the expression in the bracket: \[(x+1)^{1/2} (3x(x+1)-6) = (x+1)^{1/2} (3x^2+3x-6)\]
04

Simplify the bracket

To complete this step, the expression \(3x^2 + 3x - 6\) needs to be further simplified, which leads to: \[(x+1)^{1/2} * 3(x^2 + x - 2)\]
05

Final simplification

Breaking down the quadratic also in the bracket, obtaining: \[(x+1)^{1/2} * 3(x - 1)(x + 2)\] This simplification yields the final, simplest form of the expression.

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Key Concepts

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

Simplification of Expressions
Simplifying expressions in algebra involves making a complex expression easier to handle. It can mean factoring, expanding, or reducing it to a more straightforward form. The goal is to make calculations more manageable and to understand the expression better.

For example, in the given exercise, the expression \(3x(x+1)^{3/2}-6(x+1)^{1/2}\) needs simplification by factoring. By identifying common parts of the expression and reducing them, you turn the expression into a form that鈥檚 easier to work with.

This process often involves:
  • Finding common factors
  • Using distributive properties
  • Rewriting expressions more clearly
Remember, the key is to break down complex parts step by step.
Common Factors in Polynomials
Finding common factors in polynomials is a crucial step in the simplification process. A common factor is a term that is present in each part of an expression.

In the given exercise, both terms, \(3x(x+1)^{3/2}\) and \(-6(x+1)^{1/2}\), share \((x+1)^{1/2}\) as a common factor. Recognizing this helps you factor it out, which simplifies the entire expression.

To identify common factors:
  • Look for terms that appear in multiple parts of the expression.
  • Higher powers often indicate common factors needing factoring.
  • Extract and simplify based on common components.
This approach makes polynomial expressions easier to solve.
Quadratic Equations
Quadratic equations are polynomial equations of the form \(ax^2 + bx + c\). These can often be factored further to find solutions or simplify expressions.

During the simplification process in the exercise, the expression \(3x^2 + 3x - 6\) appears inside the brackets. This is a quadratic equation that needs further breaking down.

Simplifying quadratics includes:
  • Factoring into binomials when possible.
  • Using identities or the quadratic formula if needed.
  • Recognizing patterns like difference of squares.
In the provided solution, the expression becomes \(3(x - 1)(x + 2)\), showing a clear factorization of the quadratic. Understanding these steps is vital for solving various algebra problems.

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