Subtracting polynomials is quite similar to adding polynomials, but there are those pesky "minus" signs to deal with. If the subtraction is being done horizontally, then the "minus" signs will need to be taken carefully through the parentheses. If the subtraction is done vertically, then all that's needed is flipping all of the subtracted polynomial's signs to their opposites.
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The first thing I have to do is take that "minus" sign through the parentheses containing the second polynomial. Some students find it helpful to put a "1" in front of the parentheses, to help them keep track of the minus sign.
Here's what the subtraction looks like, when working horizontally:
(x^{3} + 3x^{2} + 5x – 4) – (3x^{3} – 8x^{2} – 5x + 6)
(x^{3} + 3x^{2} + 5x – 4) – 1(3x^{3} – 8x^{2} – 5x + 6)
(x^{3} + 3x^{2} + 5x – 4) – 1(3x^{3}) – 1 (–8x^{2}) – 1(–5x) – 1(6)
x^{3} + 3x^{2} + 5x – 4 – 3x^{3} + 8x^{2} + 5x – 6
x^{3} – 3x^{3} + 3x^{2} + 8x^{2} + 5x + 5x – 4 – 6
–2x^{3} + 11x^{2} + 10x –10
And here's what the subtraction looks like, when going vertically:
In the horizontal addition (above), you may have noticed that running the negative through the parentheses changed the sign on each and every term inside those parentheses. The shortcut when working vertically is to not bother writing in the subtaction sign or the parentheses; instead, write the second polynomial in the second row, and then just flip all the signs in that row, "plus" to "minus" and "minus" to "plus".
I'll change all the signs in the second row (shown in red below), and add down:
Either way, I get the answer:
–2x^{3} + 11x^{2} + 10x – 10
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Here's the subtraction, done horizontally:
(6x^{3} – 2x^{2} + 8x) – (4x^{3} – 11x + 10)
(6x^{3} – 2x^{2} + 8x) – 1(4x^{3} – 11x + 10)
(6x^{3} – 2x^{2} + 8x) – 1(4x^{3}) – 1(–11x) – 1(10)
6x^{3} – 2x^{2} + 8x – 4x^{3} + 11x – 10
6x^{3} – 4x^{3} – 2x^{2} + 8x + 11x – 10
2x^{3} – 2x^{2} + 19x – 10
Going vertically, I'll write out the polynomials, leaving gaps as necessary:
Then I'll flip all of the signs in the second line, and then add down:
Either way, I get the same answer:
2x^{3} – 2x^{2} + 19x – 10
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Are we limited to only adding or subtracting pairs of polynomials? No, not at all. Especially once you get to calculus, it is very likely that it will be necessary to combine three or more polynomials, some of which are added and others which are subtracted. Just take care to write things out neatly, and don't try to do too much in any one step.
Okay; to make this easier on myself, I'm first going to flip all of the signs for the second parenthetical, because there's currently a "minus" sign in front of that polynomial. So that middle polynomial becomes:
–x^{3} – 2x^{2} – 4
Then I'll set up my simplification (which now involves only addition) in the vertical format:
Then my hand-in answer is:
8x^{3} + 10x^{2} – 8x + 1
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