Mastering the Process of Adding Polynomials with Ease

a In order to add the two given polynomials, rearrange the terms so that like terms are next to each other. Then, add the like terms and simplify the sum.

(- 0.023 n^{3} + 0.08 n^{2} + 0.1 n + 15) + (- 0.41 n^{2} + 1.3 n + 36)

RemovePar

Remove parentheses

- 0.023 n^{3} + 0.08 n^{2} + 0.1 n + 15 - 0.41 n^{2} + 1.3 n + 36

CommutativePropAdd

\CommutativePropAdd

- 0.023 n^{3} + (0.08 n^{2} - 0.41 n^{2}) + (0.1 n + 1.3 n) + (15 + 36)

AddSubTerms

Add and subtract terms

- 0.023 n^{3} + (- 0.33 n^{2}) + 1.4 n + 51

AddNeg

$a + (- b) = a - b$

- 0.023 n^{3} - 0.33 n^{2} + 1.4 n + 51

Note that the polynomial is written in standard form as the degrees of its monomials decrease moving from left to right.

b To find the sum of

R + B,

the polynomials will be added using the vertical format of polynomial addition. Again, the sum can be found by adding the like terms of the polynomials. To perform the addition correctly, do not forget to align the like terms vertically.

l a l a l a l a a (- + (- 0.023 n^{3} + (- 0.023 n^{3} - 0.41 n^{2} + 1.3 n + 36 0.08 n^{2} + 0.1 n + 15) 0.33 n^{2} + 1.4 n + 51

c Now, the sums found in Parts A and B can be compared.

B + R = - 0.023 n^{3} - 0.33 n^{2} + 1.4 n + 51 R + B = - 0.023 n^{3} - 0.33 n^{2} + 1.4 n + 51

As can be seen, these sums are the same polynomial. Recall that addition of real numbers is commutative. Since adding polynomials comes down to adding the coefficients of like terms, which are real numbers, it can be concluded that the addition of polynomials is also commutative.

d Recall that the degree of a polynomial is the highest degree of its monomials. In this case, the first monomial has the

highest

degree,

which is

3 .

- 0.023 n^{3} - 0.33 n^{2} + 1.4 n + 51

Therefore, the degree of this polynomial is

3 .

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. Since the first term has the highest degree, the

leading

coefficient

- 0.023 .

- 0.023 n^{3} - 0.33 n^{2} + 1.4 n + 51

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Challenge

Investigating Polynomial Sum and Difference

Discussion

Adding and Subtracting Polynomials

Discussion

Closure Property of Polynomial Addition and Subtraction

Example

Adding Polynomials in Different Order and Formats

Hint

Solution

Example

Subtracting Polynomials to Solve a Real-Life Problem

Hint

Solution

Pop Quiz

Determining the Degree and Leading Coefficient of Polynomials

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