You can add polynomials either in vertical form or in horizontal form.

Consider the addition of two polynomials :

(5x^{2} + 4x + 1) + (2x^{2} + 5x + 2)

In vertical form, align the like terms and add as shown below.

In horizontal form, use the Associative and Commutative Properties to regroup like terms together and combine them as shown below.

**Problem 1 :**

Add :

3x^{3} + x^{2} - 2 and 2x^{2} + 5x + 5

**Solution : **

= 3x^{3} + x^{2} - 2 + 2x^{2} + 5x + 5

Group like terms together.

= 3x^{3} + (2x^{2 }+ x^{2}) + 5x + (5 - 2)

Combine like terms.

= 3x^{3} + 3x^{2} + 5x + 3

**Problem 2 :**

Add the following polynomials :

3x - y, 2y - 2x, and x + y

**Solution : **

= (3x - y) + (2y - 2x) + (x + y)

= 3x - y + 2y - 2x + x + y

Group like terms together.

= (3x + x - 2x) + (2y + y - y)

Combine like terms.

= 2x + 2y

**Problem 3 :**

Add :

(7p^{3} + 4p^{2}- 8p + 1) and (3p^{3}- 5p^{2}- 10p + 5)

**Solution : **

= (7p^{3} + 4p^{2}- 8p + 1) + (3p^{3}- 5p^{2}- 10p + 5)

= 7p^{3} + 4p^{2}- 8p + 1 + 3p^{3}- 5p^{2}- 10p + 5

Group like terms together.

= (7p^{3} + 3p^{3}) + (4p^{2}- 5p^{2}) + (-8p - 10p) + (1 + 5)

Combine like terms.

= 10p^{3} - p^{2} -18p + 6

**Problem 4 :**

Add :

-1(x^{6} + x^{3} + 6x^{2} - 2) and 2(5x^{6} + 7x^{5} - 3x - 3)

**Solution : **

= -1(x^{6} + x^{3} + 6x^{2} - 2) + 2(5x^{6} + 7x^{5} - 3x - 3)

Distributive Property.

= -x^{6} - x^{3} - 6x^{2} + 2 + 10x^{6} + 14x^{5} - 6x - 6

Group like terms together.

= (-x^{6} + 10x^{6}) + 14x^{5} - x^{3} - 6x^{2} - 6x + (2 - 6)

Combine like terms.

= 9x^{6} + 14x^{5} - x^{3} - 6x^{2} - 6x - 4

= 6x^{7} + 10x^{7}- 2x^{6} + 6x^{6} + 4x^{4} - 3x^{3} + 2x^{3} + 2x^{2} + 2x^{2}

= 16x^{7} + 4x^{6} + 4x^{4} - x^{3} + 4x^{2}

**Problem 5 :**

The profits of two different production plants can be modeled as shown below, where x' is the number of units produced at each plant.

Eastern Plant :

-0.03x^{2} + 26x - 1600

Western Plant :

-0.02x^{2} + 20x - 1800

Use the information above to write a polynomial that represents the total profits from both plants.

**Solution :**

**Total profits from both plants :**

= Eastern plant profits + Western plant profits

= (-0.03x^{2} + 26x - 1600) + (-0.02x^{2} + 20x - 1800)

= -0.03x^{2} + 26x - 1600 - 0.02x^{2} + 20x - 1800

Group like terms together.

= (-0.03x^{2} - 0.02x^{2}) + (26x + 20x) + (-1600 - 1800)

Combine like terms.

= -0.05x^{2} + 46x - 3400

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