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
