A polynomial is an expression that consists of one or more terms with constants, variables and positive exponents. It will not be a polynomial if the variable (the x or the y, etc) is the divisor; for example, it cannot have the x (if that is the variable) as the denominator (such as `2/x` ). An example of a possible polynomial is `x^3 +2x^2+ 3` but there are many possibilities.
When dividing by polynomials, it is common to divide by one of the factors or a given factor such as (x+1) which will render a result with or without a remainder. Write it as you would the normal long division you learnt in primary school. To use the given example :
`x^3 +2x^2 +0x +3` divided by `x+1` . Note the `0x` used as there is no x but it will make a difference in subtracting and bringing down as this method is the same as long division for constants (1)divide, 2)multiply, 3)subtract, 4)bring down.)
1) `x^3` divide by `x` would render `x^2` which goes in the answer column (ie divide)
2) Multiply that result (`x^2` ) by the x (= `x^3` ) and also by the 1 from your factor of `x+1` , giving you `+x^2`
3)Subtract the result from point 2) (`x^3 + x^2` ) from the expression `x^3+2x^2+0x+3 ` which cancels out the `x^3` and leaves `x^2` as `2x^2-x^2=x^2`
4) Now bring down the `0x` and repeat from point 1 above dividing `x^2 +0x` by `x+1`
1)` x^2 divide x`` = x` which goes in the answer column (which now is therefore `x^2 +x` )
2)Multiply that result (`x)` by the `x` (from the factor `x+1` ) which =`x^2` and also multiply by the `+1` which = x
3)Subtract the result from point 2)- (`x^2+x` ) from the `x^2+0x` which cancels out the `x^2` and leaves `-x`
4) Now bring down the `+3` and repeat the process dividing` -x+3` by `x+1`
1)`-x divide x=-1` which goes into the answer column (now `x^2+x-1` )
2) Multiply the x from `x+1` by -1 which `= -x` and multiply -1 by the 1 from `x+1 = -1`
3) Subtract which cancels out the x-es and leaves +4 (3 - (-1) = 4
4) This is the remainder and the answer is` x^2+x-1 rem 4`
Always ensure that the polynomial is written in descending order; that is, from the highest power of x (if x is the variable you are using).
The synthetic method can also be used when dividing polynomials. This method uses a factor as in the previous method but applies it to the co-efficients; in this example, 1, 2, 0, 3 and as the factor is x+1, therefore x=-1
First bring down the first co-efficient as it is (ie 1)
Next, multiply the factor (-1) by this number which in this case renders -1 which you subtract from the next co-efficient (2) which =+1
Then, subtract that from the next co-efficient (0) which gives you -1
Lastly, subtract that from 3 (ie 3-(-1) = 4)
Now apply the variable eliminating the highest power (ie the third power in this case). We have co-efficients 1, 1, -1 and 4; thus = `1x^2+1x-1 rem 4` which is the same answer.
Ans: Use a factor to divide polynomials in long division.
Dividing polynomials could be as simple as the division you learned in your elementary school.
Lets take an example:
Question: dividend =`x^2 + 4x + 4` ; Divisor = (x+2)
To find: Quotient and Remainder
For the first part of the dividend polynomial i.e. `x^2`
we need to multiply the divisor (x+2) with x to get the first term matching with the first part of the dividend i .e. `x^2`
so , `(x+2) xx x = x^2 + 2x`
Quotient = x and remainder = `(x^2 + 4x + 4)-- (x^2 + 2x) = 2x + 4`
For the first term of remainder polynomial i.e.`2x` , we will multiply the divisor (x+2) with 2
So, `(x+2)xx2 = 2x + 4` , thus
Quotient = x + 2 and Remainder = `(2x + 4) - ( 2x + 4) = 0`
Thus by solving the given question `(x^2 + 4x + 4)-: (x+ 2)` we get the Remainder = 0 and quotient = (x+2)