divide p(x)by q(x) if
p(x) = x^3+31/6 x^2-2x+4
q(x) = 1+3x/2
To divide p(x) by q(x), you need to perform polynomial long division. Here's how you can do it step by step:
1. Start by setting up the division in the same format as long division, with p(x) as the dividend and q(x) as the divisor. Also, arrange the terms in descending order of degrees for both polynomials:
____________________
q(x) | p(x) (in standard form)
p(x) = x^3 + (31/6)x^2 - 2x + 4
q(x) = 1 + (3/2)x
2. Divide the first term of p(x) by the first term of q(x) to get the first term of the quotient. In this case, x^3 ÷ 1 = x^3:
x^3
3. Multiply the entire q(x) polynomial by x^3 and subtract the result from p(x). Write the subtraction result below the polynomial line:
x^3 + (31/6)x^2 - 2x + 4
- (x^3 * (1 + (3/2)x))
Result after subtraction (combining like terms):
(31/6)x^2 - (5/2)x + 4
4. Bring down the next term from p(x) (which is - (5/2)x in this case). Write it next to the subtraction result:
(31/6)x^2 - (5/2)x + 4
- (5/2)x
5. Divide the first term of the modified polynomial by the first term of q(x) to get the next term of the quotient. In this case, (31/6)x^2 ÷ 1 = (31/6)x^2:
(31/6)x^2 - (5/2)x + 4
- (5/2)x
_______________
(31/6)x^2
6. Multiply the entire q(x) polynomial by (31/6)x^2 and subtract the result from the current subtraction result. Write the subtraction result below the polynomial line:
(31/6)x^2 - (5/2)x + 4
- (5/2)x
- ((31/6)x^2 * (1 + (3/2)x))
Result after subtraction (combining like terms):
(5/2)x - (4/3)
7. Bring down the next term from p(x) (which is - (4/3) in this case). Write it next to the subtraction result:
(5/2)x - (4/3)
8. Divide the first term of the modified polynomial by the first term of q(x) to get the next term of the quotient. In this case, (5/2)x ÷ 1 = (5/2)x:
(5/2)x - (4/3)
- (5/2)x
_______________
-(4/3)
9. Multiply the entire q(x) polynomial by -(4/3) and subtract the result from the current subtraction result. Write the subtraction result below the polynomial line:
(5/2)x - (4/3)
- (5/2)x
- (-(4/3) * (1 + (3/2)x)) = (4/3) + (2/3)x
Result after subtraction (combining like terms):
-(4/3)
10. Since there are no more terms left in p(x) to bring down, we have reached the end of the division process. The quotient is the sum of the terms obtained during the division. In this case, the quotient is:
Quotient = x^3 + (31/6)x^2 - (5/2)x + (5/2)
Therefore, when you divide p(x) by q(x), the quotient is x^3 + (31/6)x^2 - (5/2)x + (5/2).
Note: These calculations were done assuming fractions and decimals, and fractions were not simplified for clarity.
just type your rational function in at
wolframalpha.com
Be sure to use parentheses. It will show how it interprets your input.