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Chapter 5: Problem 42

Subtract. $$ \left(-4 x^{2}+2 x\right)-\left(-5 x^{2}+2 x^{3}+3\right) $$

Short Answer

Expert verified
-2x^{3} + x^{2} + 2x - 3

Step by step solution

01

- Write Down the Expression

Start by writing down the given expression: \[-4x^{2} + 2x - (-5x^{2} + 2x^{3} + 3)\]
02

- Distribute the Negative Sign

Distribute the negative sign across the second set of parentheses: \[-4x^{2} + 2x - (-5x^{2}) - (2x^{3}) - (3)\] This simplifies to: \[-4x^{2} + 2x + 5x^{2} - 2x^{3} - 3\]
03

- Combine Like Terms

Combine the like terms in the expression: \[-2x^{3} + (-4x^{2} + 5x^{2}) + 2x - 3\] Which simplifies to: \[-2x^{3} + x^{2} + 2x - 3\]

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Key Concepts

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

like terms
Identifying and understanding 'like terms' is crucial in polynomial operations. Like terms are terms in an algebraic expression that have the same variable raised to the same power. For instance, in the expression \(-4x^{2} + 2x - 5x^{2} + 2x^{3} + 3\), the terms \(-4x^{2}\) and \(5x^{2}\) are like terms because both terms have the variable \(x\) raised to the power of 2. Similarly, \(2x\) and any other term with just \(x\) would be like terms.
Here are some easy steps to identify like terms:
  • Look for terms that have the exact same variable part. For example, \(7xy\) and \(3xy\) are like terms.
  • Check the exponents of each variable. For example, \(2x^{3}\) and \(x^{3}\) are like terms, but \(2x^{3}\) and \(2x^{2}\) are not.
  • Remember that constants (numbers without variables) are also like terms. So, \(3\) and \(-4\) are like terms.
distribute negative sign
Distributing the negative sign is a key step when subtracting polynomials. It ensures that each term in the polynomial is correctly adjusted to reflect the subtraction. Here's how to distribute the negative sign effectively:
When you have an expression like \(-(a + b + c)\), apply the negative sign to each term within the parentheses:
  • \(-(a + b + c) = -a - b - c\)
For the given problem, \left(-4x^{2}+2x\right)-\text{-}\text{-5x^{2}+2x^{3}+3}\, you distribute the negative sign to each term inside the second set of parentheses:
  • \-4x^{2} + 2x - \text{\text{-5x^{2}}} - \text{2x^{3}} - 3\ becomes \-4x^{2} + 2x + 5x^{2} - 2x^{3} - 3\
Distributing the negative sign changes the signs of all terms in the second polynomial, preparing them for the next step.
combine like terms
Combining like terms is a process of simplifying an algebraic expression by adding or subtracting the coefficients of like terms. This step reduces the expression to its simplest form.
Consider the expression from our example: \-4x^{2} + 2x + 5x^{2} - 2x^{3} - 3\. To combine like terms, follow these steps:
  • Group the like terms together. Here, the like terms are \-4x^{2}\ and \5x^{2}\.
  • Add or subtract the coefficients of these grouped terms:\br> - For \-4x^{2} + 5x^{2} = x^{2}\
  • The term \2x\ does not have any like terms, so it remains in the expression.
  • The same goes for the term \-2x^{3}\ and the constant \-3\.
Combining these, we get the simplified expression: \[-2x^{3} + x^{2} + 2x - 3\].
Remember, ensuring you correctly combine like terms will lead to accurate and simplified polynomial expressions.

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Most popular questions from this chapter

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=5 x-1\\\ &g(x)=2 x^{2} \end{aligned} $$

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Simplify. $$ \left[\left(5^{-3}\right)^{2}\right]^{-1} $$
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