Subtracting Polynomials
To subtract one polynomial from another, add the first polynomial to the
opposite of the polynomial being subtracted.
To find the opposite of a polynomial, multiply each term by 1.
For example:
The opposite of 5x^{2} is 5x^{2}.
The opposite of 2x + 7 is 2x  7.
Note:
Hereâ€™s a way to find the opposite of a
polynomial:
Change the sign of each term.
Example 1
Find: (18w^{2} + w  32)  (40  13w^{2})
Solution
Change the subtraction to
addition of the opposite.
Remove the parentheses.
Write like terms next to
each other.
Combine like terms. 
(18w^{2} + w  32)  (40  13w^{2})
= (18w^{2} + w  32) + (1)(40  13w^{2})
= 18w^{2} + w  32  40 + 13w^{2}
= 18w^{2} + 13w^{2} + w  32  40
= 5w^{2} + w  72 
So, (18w^{2} + w  32)  (40  13w^{2}) = 5w^{2} +
w  72
Note:We can also place one polynomial beneath
the other and subtract like terms.
To do the subtraction, we change the sign
of each term being subtracted, then add.
Example 2
Subtract (15z^{2}  5yz^{2} + 4y^{3}) from (6y^{3}
 10z^{3} + 2yz^{2}).
Solution
Be careful! â€œSubtract A from Bâ€ means B  A. The order is important.
Write the difference.
Change the subtraction to
addition of the opposite.
Remove the parentheses.
Write like terms next to
each other.
Combine like terms. 
(6y^{3}
 10z^{3} + 2yz^{2})  (15z^{2}  5yz^{2} + 4y^{3})
= (6y^{3}
 10z^{3} + 2yz^{2}) + (1) (15z^{2}  5yz^{2} + 4y^{3})
= 6y^{3}
 10z^{3} + 2yz^{2}  15z^{2} + 5yz^{2}
 4y^{3}
= 6y^{3}  4y^{3}
 10z^{3} + 2yz^{2} + 5yz^{2}  15z^{2}
= 2y^{3}
 10z^{3} + 7yz^{2}  15z^{2} 
Note:
We can also place one polynomial beneath
the other and subtract like terms.
To do the subtraction, we change the sign
of each term being subtracted, then add.
