The Horizontal and Vertical Methods

	Adding Polynomials (2x² + 3x – 7) + (x² – 4x – 8)	Subtracting Polynomials (7x² – 4x + 8) – (2x² + 3x – 5)
Step 1: Distribute	1(2x² + 3x – 7) + 1(x² – 4x – 8) = 2x² + 3x – 7 + x² – 4x – 8 Recall there is an understood “1” in front of each polynomial.	1(7x² – 4x + 8) – 1(2x² + 3x – 5) = 7x² – 4x + 8 – 2x² – 3x + 5 Notice, when you are subtracting two polynomials, you distribute a negative 1 to the second polynomial.
Step 2: Identify Like Terms	2x² + 3x – 7 + x² – 4x – 8	7x² – 4x + 8 – 2x² – 3x + 5
Step 3: Combine Like Terms	3x² – 1x – 15 Notice the value of the exponent did not change. Also, remember to pay attention to the sign of each term.	5x² – 7x + 13 Notice the value of the exponent did not change. Also, remember to pay attention to the sign of each term.

Subtract: (5x² + 8x + 1) – (2x² + 3x – 6)

Step 1: 1(5x² + 8x + 1) – 1(2x² + 3x – 6)
= 5x² + 8x + 1 – 2x² – 3x + 6

Recall there is actually an understood “1” in front of each polynomial. When subtracting two polynomials, you distribute a negative 1 to the second polynomial.

Step 2: Create and fill in a

x²’s	x’s	Constants
5x²	+ 8x	+ 1
–2x²	– 3x	+ 6

Step 3: Combine the terms in the

x²’s	x’s	Constants
5x²	+ 8x	+ 1
–2x²	– 3x	+ 6
3x²	+ 5x	+ 7

Example 2

An obtuse triangle with obtuse angle labeled as C, the upper acute angle labeled as A, and the lower acute angle labeled as B. The hypotenuse is labeled with a question mark. The lower side from B to C is labeled 3x and the other side from A to C is labeled 5x minus 1.

Marley is designing a monument with a triangular base of perimeter (9x + 5) feet. The figure shows the base with some dimensions marked.

What is a simplified expression that can be used to find the length, in feet, of side AB of the triangle?

Horizontal Method
Vertical Method

Recall that perimeter is all of the sides added together. Since the perimeter of the triangle is (9x + 5), we can subtract the sum of the other two sides from (9x + 5) to find the length of the missing side.

Perimeter – [Side AC + Side CB]

(9x + 5) – [(5x – 1) + (3x)] Set up the problem.

1(9x + 5) – [1(5x – 1) + 1(3x)] First, place the understood 1's in front of the three sets of
parentheses.

9x + 5 – [5x – 1 + 3x] Distribute to remove the parentheses.

9x + 5 – [5x – 1 + 3x] Combine like terms inside the brackets.

9x + 5 – [8x – 1]

9x + 5 – 1[8x – 1] Distribute the understood 1 to remove the brackets.

9x + 5 – 1(8x) – 1(– 1)

9x + 5 – 8x + 1

9x + 5 – 8x + 1 Identify like terms.

x + 6 Combine like terms

Set up the problem.

First, place the understood 1's in front
of the three sets of parentheses and the
bracket.

Starting with the inner parentheses,
distribute to remove the parentheses
and bracket.

Sort the like terms into columns

Combine the like terms in each column.

Write a polynomial with the combined terms.

Perimeter – [Side AC + Side CB]

(9x + 5) – [(5x – 1) + (3x)]

1(9x + 5) –1 [1(5x – 1) + 1(3x)]

9x + 5 – 5x + 1 – 3x

x terms	Constants
9x -5x -3x	5 1
x	6

x + 6