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Chapter 6: Problem 26

Simplify. $$0.8 x^{3}-7 x+2.99-1.2 x-9+0.91 x^{3}$$

Short Answer

Expert verified
Question: Simplify the polynomial expression: $$0.8x^3 - 7x + 2.99 - 1.2x - 9 + 0.91x^3$$. Answer: The simplified expression is $$1.71x^3 - 8.2x - 6.01$$.

Step by step solution

01

Identify like terms

First, we need to identify the like terms in the given expression. The expression is: $$0.8x^3 - 7x + 2.99 - 1.2x - 9 + 0.91x^3$$ The like terms are: - Cubic terms: \(0.8x^3\) and \(0.91x^3\) - Linear terms: \(-7x\) and \(-1.2x\) - Constant terms: \(2.99\) and \(-9\)
02

Add the cubic terms

Now, we'll add the cubic terms: \(0.8x^3\) and \(0.91x^3\). $$0.8x^3 + 0.91x^3 = (0.8 + 0.91)x^3 = 1.71x^3$$
03

Add the linear terms

Next, we'll add the linear terms: \(-7x\) and \(-1.2x\). $$-7x - 1.2x = (-7 -1.2)x = -8.2x$$
04

Add the constant terms

Finally, we'll add the constant terms: \(2.99\) and \(-9\). $$2.99 - 9 = -6.01$$
05

Combine the results

Now, we'll combine the simplified cubic term, linear term, and constant term that we obtained in steps 2, 3, and 4. Our simplified expression is: $$1.71x^3 - 8.2x - 6.01$$

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Key Concepts

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

Understanding Cubic Terms
In the expression you're working on, cubics are terms where the variable is raised to the power of three. They involve the form \(x^3\), which is called a cubic term. While it's a bit more complex than simpler expressions with lower powers, adding cubic terms follows the same basic rules. You simply need to find like terms with the same power and add or subtract their coefficients.
For example, in the problem, you have two cubic terms: \(0.8x^3\) and \(0.91x^3\).
  • They are like terms because they both have the \(x^3\) variable.
  • To simplify, add their coefficients: \(0.8 + 0.91\), resulting in \(1.71x^3\).
This way, you keep the variable part the same \(x^3\) and only adjust the number in front.
Identifying Linear Terms
Linear terms are components of your expression where the variable is raised to the power of one, such as \(x\), which doesn't have an exponent written out but is understood to be one. Linear equations are the most basic types of polynomial equations, and they form simple lines when graphed. In the given problem, you're dealing with linear terms like \(-7x\) and \(-1.2x\).
  • These terms are easy to combine because they both have the same variable \(x\).
  • Add or subtract their coefficients: \(-7 + (-1.2) = -8.2\).
The result is \(-8.2x\), which simplifies your original expression by effectively combining these similar terms into one concise term.
Simplifying Constant Terms
Constant terms in an expression are those without any variable factor. They do not change regardless of the value of \(x\) because they lack any variable factors. In polynomial expressions, you will often encounter these numbers as a part of the sequence. Constant terms in our simplified exercise are \(2.99\) and \(-9\).
  • You simply need to combine these numbers by straightforward addition or subtraction.
  • In this instance, you subtract \(9\) from \(2.99\), leading to \(-6.01\).
This results in a simplified expression containing no variables, which we simply refer to as a constant term. Combining these constant terms correctly is crucial for reaching the most simplified form of your polynomial expression.

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