Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x)/D(x) in the form
P(x)
D(x)
= Q(x) +
R(x)
D(x)
.
P(x) = 10x3 + x2 − 21x + 9, D(x) = 5x − 7
p(x)/d(x)=
you can see all the details of long division by entering your polynomials here:
http://calc101.com/webMathematica/long-divide.jsp
To divide the polynomial P(x) = 10x^3 + x^2 - 21x + 9 by the polynomial D(x) = 5x - 7, you can use long division. Here's how you can do it:
Step 1: Write the polynomials in descending order of their powers:
P(x) = 10x^3 + x^2 - 21x + 9
D(x) = 5x - 7
Step 2: Divide the highest power term of P(x) (10x^3) by the highest power term of D(x) (5x). The result is the first term of the quotient Q(x):
Q(x) = 2x^2
Step 3: Multiply the entire D(x) by the first term of the quotient Q(x) (2x^2):
2x^2 * (5x - 7) = 10x^3 - 14x^2
Step 4: Subtract the result from P(x):
P(x) - (10x^3 - 14x^2) = 11x^2 - 21x + 9
Step 5: Bring down the next term of P(x) (-21x) and repeat steps 2-4:
Q(x) = 2x^2 - 4
2x^2 - 4 * (5x - 7) = 10x^2 - 24
P(x) - (10x^2 - 24) = -21x + 33
Step 6: Repeat steps 2-5 with the new polynomial (-21x + 33):
Q(x) = 2x^2 - 4 - 4x
(2x^2 - 4 - 4x) * (5x - 7) = 10x^2 - 14x - 28
P(x) - (10x^2 - 14x - 28) = -7x + 61
Step 7: Since -7x is of a lower degree than D(x), it's the remainder R(x).
Putting it all together, the quotient P(x)/D(x) can be expressed as:
P(x)/D(x) = Q(x) + R(x)/D(x) = 2x^2 - 4 - 4x + (-7x + 61)/(5x - 7)