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Chapter 10: Problem 39

Add \(\left(-12 x^{4}-2 x^{2}+6 x\right)\) to \(\left(9 x^{4}+2 x^{3}+4 x\right)\).

Short Answer

Expert verified
-3x^4 + 2x^3 - 2x^2 + 10x

Step by step solution

01

Identify the given polynomials

The polynomials to be added are \((-12 x^{4}-2 x^{2}+6 x)\) and \(9 x^{4}+2 x^{3}+4 x\).
02

Align like terms

Write the polynomials in a way that aligns like terms. This looks like: \(-12x^4 \; -2x^2 \; +6x\) and \(+9x^4 \; +2x^3 \; +4x\).
03

Combine the coefficients of like terms

Add together the coefficients of like terms: \((-12x^4 + 9x^4) + (2x^3) + (-2x^2) + (6x + 4x)\).
04

Simplify each term

Perform the arithmetic for each set of like terms: \(-12x^4 + 9x^4 = -3x^4, \ 2x^3 = 2x^3, \ -2x^2 = -2x^2, \ 6x + 4x = 10x\).
05

Write the simplified polynomial

Combine the simplified terms to get the final polynomial: \(-3x^4 + 2x^3 - 2x^2 + 10x\).

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Key Concepts

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

like terms
Like terms are terms in a polynomial that have the exact same variable raised to the same power. For example, in the polynomials \(-12x^4 - 2x^2 + 6x\) and \(9x^4 + 2x^3 + 4x\), \(x^4\) terms (like \(-12x^4\) and \(9x^4\)) are like terms because they both have the variable \(x\) raised to the power of 4. Recognizing and aligning these like terms is the first step in polynomial addition. This helps to simplify the process by focusing on combining only the coefficients of like terms.
combine coefficients
Once the like terms are identified, the next step is to combine the coefficients of these terms. Coefficients are the numerical parts of the terms. For instance, in the term \(-12x^4\), \(-12\) is the coefficient, and for \(9x^4\), \(9\) is the coefficient. When adding polynomials, you combine the coefficients of the like terms while keeping the variable part unchanged. So, \(-12x^4\) combined with \(9x^4\) becomes \(-12 + 9 = -3\), resulting in \(-3x^4\). Similarly, you add \((6 + 4)\) to combine the coefficients of \(6x\) and \(4x\), which results in \(10x\).
simplifying polynomials
After combining the coefficients of the like terms, you simplify the polynomial by performing the arithmetic operations. In our example, the simplified terms are \(-3x^4\), \(2x^3\), \(-2x^2\), and \(+10x\). These simplified terms are then combined to form the final polynomial: \(-3x^4 + 2x^3 - 2x^2 + 10x\). Simplifying the polynomial means ensuring there are no more terms that can be combined further, and that it is written in standard form. This typically means writing it with the highest degree term first, then the next highest, and so on, down to the constant (if there is one). By doing this, we make the polynomial easier to understand and work with for future calculations.

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Most popular questions from this chapter

Subtract the polynomials. $$\left(-8 y^{4}+2 y^{2}-3 y+10\right)-\left(6 y^{4}+y^{3}+11 y-9\right)$$

The cost (in dollars) for a speeding ticket is given by the polynomial \(110+15 x .\) In this context, \(x\) is the number of miles per hour a motorist travels over the speed limit. a. Evaluate the polynomial for \(x=15\) and interpret the answer in the context of this problem. b. Evaluate the polynomial for \(x=25\) and interpret the answer in the context of this problem.

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