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Chapter 5: Problem 95

Simplify. $$ \left(345.099 x^{3}-6.178 x\right)-\left(94.508 x^{3}-8.99 x\right) $$

Short Answer

Expert verified
250.591 x^3 - 15.168 x

Step by step solution

01

Identify like terms

Identify the terms involving the same powers of x from both expressions. The like terms here are the terms with \(x^3\) and the linear terms with \(x\).
02

Group like terms

Rewrite the expression grouping the like terms together:\[\left(345.099 x^3-94.508 x^3\right)-\left(6.178 x - 8.99 x\right)\]
03

Simplify each group

Simplify each set of like terms separately. For the \(x^3\) terms:\[345.099 x^3 - 94.508 x^3 = (345.099 - 94.508) x^3 = 250.591 x^3\]For the linear \(x\) terms:\[-6.178 x + 8.99 x = -6.178 x - (-8.99 x) = (6.18 - 8.99)x = -15.168 x\]
04

Combine simplified terms

Combine all the simplified terms to get the final expression:\[250.591 x^3 - 15.168 x\]

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Key Concepts

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

like terms
Understanding 'like terms' is the first step to simplify polynomials. Like terms are terms that have the same variables raised to the same power.
For example, in the expression \(3x^2 + 4x^2 + x\), the terms \(3x^2\) and \(4x^2\) are like terms because they both contain \(x^2\).
However, \(3x\) and \(4x^2\) are not like terms because the exponents of \(x\) are different.
To simplify an expression, we group all the like terms together to make combining them easier.
polynomial subtraction
Subtraction of polynomials is very similar to subtracting regular numbers. However, it also involves distributing the subtraction sign and then combining the like terms.
This is done by subtracting the coefficients (the numerical part) of the like terms.
For example, if we have \( (7x^2 - x) - (3x^2 + 2x)\), we first rewrite this as \[ 7x^2 - x - 3x^2 - 2x \]
Next, we group the like terms together: \[ (7x^2 - 3x^2) + (-x - 2x) \]
After that, we subtract the coefficients: \[ 4x^2 - 3x \]
algebraic expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols, but no equal signs.
Examples of algebraic expressions include: \(3x + 2\), \(5a^2 - 3b + 7\), and \(2x^3 + 4y^2 - x + 8\).
In these expressions, \(x\) and \(y\) are variables, while \(3, 2, 5, 7, 4, 8\) are constants.
Algebraic expressions can be combined, simplified, or factored using various mathematical operations.
simplifying polynomials
Simplifying polynomials involves reducing them to their simplest form by combining all like terms.
This process makes the polynomial easier to work with in further calculations.
Following the steps from our exercise, we:
  • Identified the like terms \(345.099 x^3 - 94.508 x^3\) and \(-6.178 x - (-8.99) x\)
  • Grouped the like terms: \[ (345.099 x^3 - 94.508 x^3) - (6.178 x - 8.99 x) \]
  • Simplified each group separately: \[ 345.099 x^3 - 94.508 x^3 = 250.591 x^3\
    and
    \ -6.178 x + 8.99 x = -15.168 x \]
  • Combined the simplified terms to get the final expression: \[ 250.591 x^3 - 15.168 x \]

The key is to always simplify each set of like terms first, then combine everything at the end.

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