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Addition of Polynomials

To add two polynomials, combine their like terms.

Example 1: (4x5 + x4 -12x3 + x - 6) + (3x4 +8x3 +6x2 - x) =?

= 4x5 + (1 + 3)x4 + (- 12 + 8)x3 +6x2 + (- 1 + 1)x - 6
= 4x5 +4x4 -4x3 +6x2 + 0x - 6
= 4x5 +4x4 -4x3 +6x2 - 6

Example 2: (- 3x4 -2x2 + x + 17) + (14x6 +12x5 -2x4 +7x3 +2x2 - 4x - 12) =?

= 14x6 +12x5 + (- 3 - 2)x4 +7x3 + (- 2 + 2)x2 + (1 - 4)x + (17 - 12)
= 14x6 +12x5 -5x4 +7x3 +0x2 - 3x + 5
= 14x6 +12x5 -5x4 +7x3 - 3x + 5

Subtraction of Polynomials

To subtract one polynomial from another, change the subtraction sign to an addition sign and change the signs of all the terms in the polynomial being subtracted (don't forget to change the sign of the first term). Then add the resulting polynomial by combining like terms.

Example 3: (3x4 +4x3 -5x2 +18) - (7x5 +3x4 -12x3 -3x2 + 7x - 9) =?
= (3x4 +4x3 -5x2 +18) + (- 7x5 -3x4 +12x3 +3x2 - 7x + 9)
= - 7x5 + (3 - 3)x4 + (4 + 12)x3 + (- 5 + 3)x2 - 7x + (18 + 9)
= - 7x5 +16x3 -2x2 - 7x + 27

Example 4: (- 5x6 -3x4 +18x3 -7x2 +9x + 11) - (- x4 + x3 -12x2 + 10x + 5) =?
= (- 5x6 -3x4 +18x3 -7x2 +9x + 11) + (x4 - x3 +12x2 - 10x - 5)
= - 5x6 + (- 3 + 1)x4 + (18 - 1)x3 + (- 7 + 12)x2 + (9 - 10)x + (11 - 5)
= - 5x6 -2x4 +17x3 +5x2 - x + 6

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