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) = 6x4 − 3x3 + 29x2, D(x) = 3x2 + 13
*************_2x^2___________-1x+1
(3x^2 + 13) / 6x^4-3x^3+29x^2+0x+0
**************6x^4_____+26x^2_____
*******************-3x^3+3x^2 +0x+0
*******************-3x^3______-13x
************************ +3x^2+13x+0
************************ +2x^2____+13
******************************+13x-13
so
2x^2-x+1 and (13x-13)/(3x^2+13)
To divide P(x) = 6x^4 - 3x^3 + 29x^2 by D(x) = 3x^2 + 13, we can use long division:
2x^2 - x + 6
-------------------------
3x^2 + 13 | 6x^4 - 3x^3 + 29x^2 + 0x + 0
6x^4 + 26x^2
--------------
3x^2 +3x
3x^2 + 13
---------
-4x
The result of the division is
P(x)/D(x) = 2x^2 - x + 6 - (4x / (3x^2 + 13))
So, the quotient Q(x) is 2x^2 - x + 6, and the remainder R(x) is -4x.
To divide P(x) by D(x) using synthetic division, follow these steps:
Step 1: Set up the division in the proper format:
Write the polynomials in descending order of the exponent of x, leaving the powers with no terms represented as 0. In this case, we have:
P(x) = 6x^4 - 3x^3 + 29x^2 + 0x + 0
D(x) = 3x^2 + 0x + 13
Step 2: Identify the leading term of the divisor (D(x)), which is 3x^2.
Step 3: Divide the leading term of the dividend (P(x)) by the leading term of the divisor:
6x^4 / 3x^2 = 2x^2
Step 4: Multiply the divisor (D(x)) by the quotient obtained in Step 3. Place this product underneath the dividend, lined up by their respective exponents:
2x^2 * (3x^2 + 0x + 13) = 6x^4 + 0x^3 + 26x^2
Step 5: Subtract the result obtained in Step 4 from the original dividend (P(x)). Place the result of the subtraction below the line:
(6x^4 - 3x^3 + 29x^2) - (6x^4 + 0x^3 + 26x^2) = -3x^3 + 3x^2
Step 6: Repeat the process with the new dividend:
-3x^3 / 3x^2 = -x
Step 7: Multiply the divisor (D(x)) by the quotient obtained in Step 6. Place this product underneath the new dividend, lined up by their respective exponents:
-x * (3x^2 + 0x + 13) = -3x^3 + 0x^2 - 13x
Step 8: Subtract the result obtained in Step 7 from the previous result:
(-3x^3 + 3x^2) - (-3x^3 + 0x^2 - 13x) = 3x^2 + 13x
Step 9: Repeat the process with the new dividend:
(3x^2 + 13x) / (3x^2 + 13)
Since the degree of the new dividend is less than the degree of the divisor, we cannot divide any further. Therefore, the quotient (Q(x)) is 2x^2 - x and the remainder (R(x)) is 3x^2 + 13x.
Expressing P(x)/D(x) in the desired form:
P(x)D(x) = Q(x) + R(x)D(x)
6x^4 - 3x^3 + 29x^2 = (2x^2 - x) (3x^2 + 13) + (3x^2 + 13x) (3x^2 + 13)
Thus, P(x)/D(x) can be expressed as: 2x^2 - x + (3x^2 + 13x) (3x^2 + 13)/(3x^2 + 13)