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Subtracting Polynomials (page 2 of 2)

Subtracting polynomials is quite similar to adding polynomials, but you have that pesky minus sign to deal with. Here are some examples, done both horizontally and vertically:

• Simplify (x3 + 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6)

The first thing I have to do is take that negative through the parentheses. Some students find it helpful to put a "1" in front of the parentheses, to help them keep track of the minus sign:

Horizontally:

(x3 + 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6)
=  (x3 + 3x2 + 5x – 4) –
1(3x3 – 8x2 – 5x + 6)
=  (x3 + 3x2 + 5x – 4) – 1(3x3) – 1 (–8x2) – 1(–5x) – 1(6)
=  x3 + 3x2 + 5x – 4 – 3x3 + 8x2 + 5x – 6

=  x3 – 3x3 + 3x2 + 8x2 + 5x + 5x – 4 – 6

=
–2x3 + 11x2 + 10x –10

In the horizontal case, you may have noticed that running the negative through the parentheses changed the sign on each term inside the parentheses. The shortcut here is to not bother writing in the subtaction sign or the parentheses; instead, you just change all the signs in the second row.

I'll change all the signs in the second row (shown in red below), and add down:

Either way, I get the answer:  –2x3 + 11x2 + 10x – 10

• Simplify (6x3 – 2x2 + 8x) – (4x3 – 11x + 10)

Horizontally:

(6x3 – 2x2 + 8x) – (4x3 – 11x + 10)
=  (6x3 – 2x2 + 8x) –
1(4x3 – 11x + 10)
=  (6x3 – 2x2 + 8x) – 1(4x3) – 1(–11x) – 1(10)
=  6x3 – 2x2 + 8x – 4x3 + 11x – 10

=  6x3 – 4x3 – 2x2 + 8x + 11x – 10

=
2x3 – 2x2 + 19x – 10

Vertically: I'll write out the polynomials, leaving gaps as necessary:

Then I'll change the signs in the second line, and add:

Either way, I get the answer: 2x3 – 2x2 + 19x – 10

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