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# 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 5x2 is -5x2.

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: (-18w2 + w - 32) - (40 - 13w2)

 Solution Change the subtraction to addition of the opposite. Remove the parentheses. Write like terms next to each other. Combine like terms. (-18w2 + w - 32) - (40 - 13w2) = (-18w2 + w - 32) + (-1)(40 - 13w2) = -18w2 + w - 32 - 40 + 13w2 = -18w2 + 13w2 + w - 32 - 40 = -5w2 + w - 72

So, (-18w2 + w - 32) - (40 - 13w2) = -5w2 + 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 (15z2 - 5yz2 + 4y3) from (6y3 - 10z3 + 2yz2).

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. (6y3 - 10z3 + 2yz2) - (15z2 - 5yz2 + 4y3) = (6y3 - 10z3 + 2yz2) + (-1) (15z2 - 5yz2 + 4y3) = 6y3 - 10z3 + 2yz2 - 15z2 + 5yz2 - 4y3 = 6y3  - 4y3 - 10z3 + 2yz2 + 5yz2 - 15z2 = 2y3 - 10z3 + 7yz2 - 15z2

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.