6. In each of the following, divide
f x
by
dx
, obtaining quotient
qx
and remainder
rx.
Then, write a division statement, i.e. express your answers in the form
f x dxqx rx.
a)
3 2 2
3 2
x x x x
b)
2 4 3 5 3
3 2
x x x x
c)
3 4 4
2
x x d)
4 6 6 9 2 3
3 2
x x x x
e)
3 7 5 1 3 1
3 2
x x x x
f)
2 3 8 4
4 3 2
x x x x x
g)
1 1
a)
Dividing (x^3 + 2x^2 + x - 2) by (3x^2 - x + 2):
q(x) = (1/3)x - (7/9)
r(x) = -(2/9)x + (16/9)
Division statement: (x^3 + 2x^2 + x - 2) = (3x^2 - x + 2)(1/3)x - (7/9) + ( -(2/9)x + (16/9) )
b)
Dividing (2x^4 + 3x^3 - x^2 - 5) by (3x^2 - x - 1):
q(x) = (2/3)x^2 + (1/3)x + (1/9)
r(x) = -(8/9)x + (4/9)
Division statement: (2x^4 + 3x^3 - x^2 - 5) = (3x^2 - x - 1)(2/3)x^2 + (1/3)x + (1/9) + ( -(8/9)x + (4/9) )
c)
Dividing (3x^4 - 2x^2 + 4) by (4x^2):
q(x) = (3/4)x^2 - (2/16)
r(x) = 0
Division statement: (3x^4 - 2x^2 + 4) = (4x^2)(3/4)x^2 - (2/16)
d)
Dividing (4x^3 + 6x^2 - 6x - 9) by (2x - 3):
q(x) = 2x^2 + 6
r(x) = 3
Division statement: (4x^3 + 6x^2 - 6x - 9) = (2x - 3)(2x^2 + 6) + 3
e)
Dividing (3x^3 + 7x^2 + 5x + 1) by (3x + 1):
q(x) = x^2 + 2
r(x) = -1
Division statement: (3x^3 + 7x^2 + 5x + 1) = (3x + 1)(x^2 + 2) - 1
f)
Dividing (2x^3 + 3x^2 - 8) by (4x^3 - x^2 + x - 2):
q(x) = (1/4)
r(x) = 0
Division statement: (2x^3 + 3x^2 - 8) = (4x^3 - x^2 + x - 2)(1/4)
g)
Dividing (x - 1) by (x - 1):
q(x) = 1
r(x) = 0
Division statement: (x - 1) = (x - 1)(1) + 0
To divide two polynomials and obtain the quotient and remainder, follow these steps:
Step 1: Write the polynomial division statement in the form f(x) = d(x) * q(x) + r(x), where f(x) is the dividend, d(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.
Step 2: Identify the degree of the divisor and the dividend. The degree is the highest exponent of the variable.
Step 3: Divide the term with the highest degree in the dividend by the term with the highest degree in the divisor. This gives you the first term of the quotient q(x).
Step 4: Multiply the entire divisor by the first term of the quotient q(x), and subtract it from the dividend. This gives you a new polynomial.
Step 5: Repeat steps 3 and 4 with the new polynomial as the dividend until you cannot divide further.
Step 6: The remaining polynomial after division is the remainder r(x).
Now, let's apply these steps to each problem one by one:
a) Divide (3x^2 - x + 2) by (2x - 3):
Step 1: f(x) = 3x^2 - x + 2, d(x) = 2x - 3, q(x) = ?, r(x) = ?
Step 2: The degree of the dividend is 2 and the degree of the divisor is 1.
Step 3: Divide the highest degree terms: (3x^2 / 2x) = (3/2)x.
Step 4: Multiply (2x - 3) by (3/2)x: (3/2)x * (2x - 3) = 3x^2 - (9/2)x.
Step 5: Subtract the result from the dividend: (3x^2 - x + 2) - (3x^2 - (9/2)x) = (7/2)x + 2.
Step 6: The remainder is (7/2)x + 2.
Therefore, the division statement is: 3x^2 - x + 2 = (2x - 3)*(3/2)x + (7/2)x + 2.
Repeat the steps for the remaining problems (b) to (g) following the same process.
a) To divide f(x) = (3x^2 - x + 2) by d(x) = (2x), we use long division:
______________
2x | 3x^2 - x + 2
- (3x^2 - x)
_____________
0x + 2
Therefore, the quotient q(x) = 0 and the remainder r(x) = 2.
The division statement is f(x) = d(x) * q(x) + r(x), so in this case it is:
f(x) = (2x) * 0 + 2.
b) To divide f(x) = (2x^4 - 3x^3 - x^2 + 5) by d(x) = (3x^2 - x), using long division:
________
3x^2 - x | 2x^4 - 3x^3 - x^2 + 5
- (2x^4 - x^3)
______________
-2x^3 - x^2
+ (2x^3 - x^2)
______________
0
The quotient q(x) = 0 and the remainder r(x) = 0.
The division statement is f(x) = d(x) * q(x) + r(x), so in this case it is:
f(x) = (3x^2 - x) * 0 + 0.
c) To divide f(x) = (3x^4 - x^2) by d(x) = (4x^2), we use long division:
_________
4x^2 | 3x^4 - x^2
- (3x^4 - x^2)
_______________
0
The quotient q(x) = 0 and the remainder r(x) = 0.
The division statement is f(x) = d(x) * q(x) + r(x), so in this case it is:
f(x) = (4x^2) * 0 + 0.
d) To divide f(x) = (4x^6 + 6x^3 - 2x^2 + 9) by d(x) = (2x^3 + x^2), using long division:
__________________
2x^3 + x^2 | 4x^6 + 6x^3 - 2x^2 + 9
- (4x^6 + 2x^5)
___________________
- 2x^5 - 2x^2 + 9
+ (2x^5 + x^4)
___________________
- x^4 - 2x^2 + 9
+ (x^4 + 0x^3)
___________________
- 2x^2 + 9
+ (2x^2 + 0x)
___________________
9
Therefore, the quotient q(x) = -2x^3 + x^4 and the remainder r(x) = 9.
The division statement is f(x) = d(x) * q(x) + r(x), so in this case it is:
f(x) = (2x^3 + x^2) * (-2x^3 + x^4) + 9.
e) To divide f(x) = (3x^7 + 5x^3 + x + 7) by d(x) = (3x^2 + x), using long division:
__________________
3x^2 + x | 3x^7 + 5x^3 + x + 7
- (3x^7 + x^5)
____________________
0x^5 + 5x^3 + x + 7
+ (0x^5 + 0x^4)
_____________________
5x^3 + x + 7
- (5x^3 + x^2)
_____________________
- x^2 + x + 7
+ (0x^2 + 0x)
_____________________
- x^2 + x + 7
- ( - x^2 - 0x^2)
_____________________
2x + 7
Therefore, the quotient q(x) = 0x^5 + 5x^3 - x^2 + x + 7 and the remainder r(x) = 2x + 7.
The division statement is f(x) = d(x) * q(x) + r(x), so in this case it is:
f(x) = (3x^2 + x) * (0x^5 + 5x^3 - x^2 + x + 7) + (2x + 7).
f) To divide f(x) = (2x^4 - 3x^3 + 8x) by d(x) = (4x^3 - x^2 + x - 2), using long division:
________________
4x^3 - x^2 + x - 2 | 2x^4 - 3x^3 + 8x
- (2x^4 - x^3 + x^2 - 2x)
_________________________
- 2x^3 + 9x
+ (2x^3 - 9x^2 + 9x - 18)
_________________________
- 9x^2 + 17x - 18
+ ( - 9x^2 + 9x - 9)
___________________________
26x - 27
Therefore, the quotient q(x) = -2x + 2x^2 and the remainder r(x) = 26x - 27.
The division statement is f(x) = d(x) * q(x) + r(x), so in this case it is:
f(x) = (4x^3 - x^2 + x - 2) * (-2x + 2x^2) + (26x - 27).
g) To divide f(x) = (x) by d(x) = (1), the division is straightforward:
f(x) = (1) * (x) + 0.
Therefore, the quotient q(x) = x and the remainder r(x) = 0.
The division statement is f(x) = d(x) * q(x) + 0, which can be simplified to f(x) = d(x) * q(x).