Adding and Subtracting Polynomials

The procedure used in the addition and subtraction of polynomials involves techniques for reducing similar terms, sign play, operations involving equal signs and different signs. Note the following examples:
Addition
Example 1
Add x² – 3x – 1 with –3x² + 8x – 6.
(x² – 3x – 1) + (–3x² + 8x – 6) → eliminate the second parentheses through sign play.
+(–3x²) = -3x²
+(+8x) = +8x
+(–6) = –6
x² – 3x – 1 –3x² + 8x – 6 → reduce similar terms.
x² – 3x² – 3x + 8x – 1 – 6
-2x² + 5x – 7
Therefore: (x² – 3x – 1) + (–3x² + 8x – 6) = –2x² + 5x – 7
Example 2
Adding 4x² – 10x – 5 and 6x + 12, we will have:
(4x² – 10x – 5) + (6x + 12) → eliminate parentheses using sign set.
4x² – 10x – 5 + 6x + 12 → reduce similar terms.
4x² – 10x + 6x – 5 + 12
4x² – 4x + 7
Therefore: (4x² – 10x – 5) + (6x + 12) = 4x² – 4x + 7
Subtraction
Example 3
Subtracting –3x² + 10x - 6 out of 5x² – 9x – 8.
(5x² – 9x – 8) – (-3x² + 10x – 6) → remove parentheses using sign set.
– (-3x²) = +3x²
– (+10x) = –10x
– (–6) = +6
5x² – 9x – 8 + 3x² –10x +6 → reduce similar terms.

5x² + 3x² – 9x –10x – 8 + 6
8x² – 19x – 2
Therefore: (5x² – 9x – 8) – (-3x² + 10x – 6) = 8x² – 19x – 2
Example 4
If we subtract 2x³ - 5x² - x + 21 and 2x³ + x² - 2x + 5, we have:
(2x³ – 5x² – x + 21) – (2x³ + x² – 2x + 5) → eliminating the parentheses through the game of signs.
2x³ – 5x² – x + 21 – 2x³ – x² + 2x – 5 → reduction of similar terms.
2x³ - 2x³ - 5x² - x² - x + 2x + 21 - 5
0x³ - 6x² + x + 16
– 6x² + x + 16
Therefore: (2x³ - 5x² - x + 21) - (2x³ + x² - 2x + 5) = - 6x² + x + 16
Example 5
Considering the polynomials A = 6x³ + 5x² – 8x + 15, B = 2x³ – 6x² – 9x + 10 and C = x³ + 7x² + 9x + 20. Calculate:
a) A + B + C
(6x³ + 5x² - 8x + 15) + (2x³ - 6x² - 9x + 10) + (x³ + 7x² + 9x + 20)
6x³ + 5x² - 8x + 15 + 2x³ - 6x² - 9x + 10 + x³ + 7x² + 9x + 20
6x³ + 2x³ + x³ + 5x² - 6x² + 7x² - 8x - 9x + 9x + 15 + 10 + 20
9x³ + 6x² - 8x + 45
A + B + C = 9x³ + 6x² - 8x + 45
b) A - B - C
(6x³ + 5x² - 8x + 15) - (2x³ - 6x² - 9x + 10) - (x³ + 7x² + 9x + 20)
6x³ + 5x² - 8x + 15 - 2x³ + 6x² + 9x - 10 - x³ - 7x² - 9x - 20
6x³ - 2x³ - x³ + 5x² + 6x² - 7x² - 8x + 9x - 9x + 15 - 10 - 20
6x³ - 3x³ + 11x² - 7x² - 17x + 9x + 15 - 30
3x³ + 4x² – 8x – 15
A - B - C = 3x³ + 4x² - 8x - 15

by Mark Noah
Graduated in Mathematics
Brazil School Team

Polynomials - Math - Brazil School

Source: Brazil School - https://brasilescola.uol.com.br/matematica/adicao-subtracao-polinomios.htm