Subtraction of polynomials is very much similar to addition of them. We can simply say that
" The subtraction of one polynomial from the second polynomial is a process of adding the second polynomial into the first polynomial with all the signs of the first polynomial inverted. "
Do not worry if you are unable to understand the above statement. We present a few examples of polynomial subtraction below, and they will make the above idea clear to you:
Here the given statement requires you to evaluate 7x - 4x which is equivalent to finding 7x + (-4x)
We can combine the two like terms through simple subtraction:
=> 7x - 4x = (7 - 4) x = 3x
Due to the sign of subtraction, we invert the signs of all the terms present inside the parentheses of the second polynomial as shown below:
`(4x^2 + xy) - (3x^2 + 7xy)`
`= 4x^2 + xy - 3x^2 - 7xy`
` = 4x^2 - 3x^2 + xy- 7xy `
`= (4 - 3)x^2 + (1 - 7)xy`
= ` x^2 - 6xy `
Again, we invert the signs of all the terms present inside the parentheses of the second polynomial while taking them out of the parentheses:
= ` 5x^3 + 3x^2y + 4xy - 6y^2 - 3x^2 - 7x^2y + 2xy - 4xy^2 + 5 `
(Combining like terms just like we did during addition)
= `5x^3 - 3x^2 + (3x^2y - 7x^2y) + (4xy + 2xy) - 6y^2 - 4xy^2 + 5`
= ` 5x^3 - 3x^2 - 4x^2y + 6xy + 2xy - 6y^2 - 4xy^2 + 5`
Hence just like addition process, only like terms can be subtracted from each other.
Just like the addition of two polynomials, we can subtract polynomials through both "vertical" and "horizontal" methods. We have presented the horizontal method above, and we include an example of vertical method below:
Using the vertical method,
We place the like terms above and below each other.
Note: In the horizontal method of polynomial subtraction, we saw that due to the -ve sign present before the 2nd polynomial, all the signs inside the parentheses got inverted. In a similar fashion, we need to invert all the signs in the lower polynomial.
`2x^3 + 4x^2 + 3x - 2`
` - 1x^3 - 5x^2 - 9x + 3`
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(Inverting the signs)
`2x^3 + 4x^2 + 3x - 2 `
` - 1x^3 + 5x^2 + 9x - 3`
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` x^3 + 9x^2 + 12x - 5 `
You may choose any of the two methods discussed above. Just make sure that you do not wrongly use the +ve and -ve signs.
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