Long Division
Long Division - How it Works - Video
Division Algorithm
Division Algorithm for Polynomials:
When we divide polynomial functions, it is just like with when we do it whole numbers. For instance, 5 / 3 = 3 * 1 + 2. Instead of whole numbers, we have different functions.
dividend = divisor * quotient + remainder => f(x) = p(x) * q(x) + r(x)
If r(x) = 0 then the divisor is a factor of the dividend.
Example 1
Example 1:
Here we have two functions f(x) = x3 - 2x2 - 5x + 6 and p(x) = x - 4. f(x) is the dividend and p(x) is the divisor.
Our first step is find a term when multiplied with x gets x3. So we need x * term = x3, and that term is x2. Once we find the term, x2, we put the term on top of the long division symbol. Then we need to multiply x2 and (x - 4) and then result x3 - 4x2 is put underneath the dividend. Now we subtract (x3 - 2x2) and (x3 - 4x2) and our result is 2x2.
Now we need to continue so we must bring down the -5x to join the +2x2. So now we have +2x2 - 5x. Our next step is find a term when multiplied with x gets 2x2. So we need x * term = 2x2, and that term is 2x. Once we find the term, 2x, we put the term on top of the long division symbol. Then we need to multiply 2x and (x - 4) and then result 2x2 - 8x is put underneath the +2x2 - 5x. Now we subtract (2x2 - 5x) and (2x2 - 8x) and our result is 3x.
Now we need to continue so we must bring down the 6 to join the +3x. So now we have +3x + 6. Our next step is find a term when multiplied with x gets 3x. So we need x * term = 3x, and that term is 3. Once we find the term, 3, we put the term on top of the long division symbol. Then we need to multiply 3 and (x - 4) and then result 3x - 12 is put underneath the +3x + 6. Now we subtract (3x + 6) and (3x - 12) and our result is 18.
So x2 + 2x + 3 is the quotient, 18 is the remainder, x3 - 2x2 - 5x + 6 is the dividend and x - 4 is the divisor.
There are two ways to write the answer. The first one is (x3 - 2x2 - 5x + 6) / (x - 4) = (x2 + 2x + 3) + 18 / (x - 4), and the second one is (x3 - 2x2 - 5x + 6) = (x2 + 2x + 3)(x - 4) + 18.
Example 2
Example 2:
Here we have two functions f(x) = x3 - 8x - 5 and p(x) = x + 2. f(x) is the dividend and p(x) is the divisor.
This time we did not have a degree 2 term or x2 with some coefficient. So we need to insert 0x2 under the long division symbol so we have the correct of columns. We want the degrees to decrease by 1.
Our next step is find a term when multiplied with x gets x3. So we need x * term = x3, and that term is x2. Once we find the term, x2, we put the term on top of the long division symbol. Then we need to multiply x2 and (x + 2) and then result x3 + 2x2 is put underneath the dividend. Now we subtract (x3 + 0x2) and (x3 + 2x2) and our result is -2x2.
Now we need to continue so we must bring down the -8x to join the -2x2. So now we have -2x2 - 8x. Our next step is find a term when multiplied with x gets -2x2. So we need x * term = -2x2, and that term is -2x. Once we find the term, -2x, we put the term on top of the long division symbol. Then we need to multiply -2x and (x + 2) and then result -2x2 - 4x is put underneath the -2x2 - 8x. Now we subtract (-2x2 - 8x) and (-2x2 - 4x) and our result is -4x.
Now we need to continue so we must bring down the -5 to join the -4x. So now we have -4x - 5. Our next step is find a term when multiplied with x gets -4x. So we need x * term = -4x, and that term is -4 . Once we find the term, -4, we put the term on top of the long division symbol. Then we need to multiply -4 and (x + 2) and then result -4x - 8 is put underneath the -4x - 5. Now we subtract (-4x - 5) and (-4x - 8) and our result is 3.
So x2 - 2x - 4 is the quotient, 3 is the remainder, x3 - 8x - 5 is the dividend and x + 2 is the divisor.
There are two ways to write the answer. The first one is (x3 - 8x - 5) / (x + 2) = (x2 - 2x - 4) + 3 / (x + 2), and the second one is (x3 - 8x - 5) = (x2 - 2x - 4)(x + 2) + 3.
Example 3
Example 3:
Here we have two functions f(x) = x3 - 3x2 - 10x + 24 and p(x) = 2x - 3. f(x) is the dividend and p(x) is the divisor.
Our first step is find a term when multiplied with 2x gets x3. So we need 2x * term = x3, and that term is (1/2)x2. Once we find the term, (1/2)x2, we put the term on top of the long division symbol. Then we need to multiply (1/2)x2 and (2x - 3) and then result x3 - (3/2)x2 is put underneath the dividend. Now we subtract (x3 - 3x2) and (x3 - (3/2)x2) and our result is (-3/2)x2.
Now we need to continue so we must bring down the -10x to join the (-3/2)x2. So now we have (-3/2)x2 - 10x. Our next step is find a term when multiplied with 2x gets (-3/2)x2. So we need 2x * term = (-3/2)x2, and that term is (-3/4)x. Once we find the term, (-3/4)x, we put the term on top of the long division symbol. Then we need to multiply (-3/4)x and (2x - 3) and then result (-3/2)x2 + (9/4)x is put underneath the (-3/2)x2 - 10x. Now we subtract (-3/2)x2 - 10x) and ((-3/2)x2 + (9/4)x) and our result is (-49/4)x.
Now we need to continue so we must bring down the 24 to join the (-49/4)x. So now we have (-49/4)x + 24. Our next step is find a term when multiplied with 2x gets (-49/4)x. So we need 2x * term = (-49/4)x, and that term is -49/8 . Once we find the term, -49/8, we put the term on top of the long division symbol. Then we need to multiply -49/8 and (2x - 3) and then result (-49/4)x + 147/8 is put underneath the (-49/4)x + 24. Now we subtract [(-49/4)x + 24] and [(-49/4)x + 147/8] and our result is 45/8.
So (1/2)x2 - (3/4)x - (49/8) is the quotient, 45/8 is the remainder, x3 - 3x2 - 10x + 24 is the dividend and 2x - 3 is the divisor.
There are two ways to write the answer. The first one is (x3 - 3x2 - 10x + 24) / (2x - 3) = (1/2)x2 - (3/4)x - (49/8) + (45/8 ) / (2x - 3), and the second one is (x3 - 3x2 - 10x + 24) = [(1/2)x2 - (3/4)x - (49/8)][(2x - 3)] + (45/8 ).