In each of the following, divide f x by dx, obtaining quotient qx and remainder rx. Then, write a division statement, i.e. express your answers in the form f x dxqx rx.

Question

In each of the following, divide f x by dx, obtaining quotient qx and remainder rx. Then, write a division statement, i.e. express your answers in the form f x dxqx rx.

x3 3x2 x2x2 3x2 4x4
3x3 7x2 5x13x1 x5 1x1
b) 2x3 4x2 3x5x3
d) 4x3 6x2 6x92x3 f) x4 x3 2x2 3x8x4

Answer 1

a) (x^3 - 3x^2 + x + 2) ÷ (x + 2) = x^2 - 5x + 11

Division statement: (x^3 - 3x^2 + x + 2) = (x + 2)(x^2 - 5x + 11)

b) (3x^2 - 4) ÷ (x - 4) = 3x + 8

Division statement: (3x^2 - 4) = (x - 4)(3x + 8)

c) (3x^3 + 7x^2 + 5x + 1) ÷ (3x + 1) = x^2 + 2x + 1

Division statement: (3x^3 + 7x^2 + 5x + 1) = (3x + 1)(x^2 + 2x + 1)

d) (x^5 - 1) ÷ (x - 1) = x^4 + x^3 + x^2 + x + 1

Division statement: (x^5 - 1) = (x - 1)(x^4 + x^3 + x^2 + x + 1)

e) (2x^3 - 4x^2 - 3x + 5) ÷ (x - 3) = 2x^2 - 10x + 27

Division statement: (2x^3 - 4x^2 - 3x + 5) = (x - 3)(2x^2 - 10x + 27)

f) (4x^3 + 6x^2 - 6x - 9) ÷ (2x + 3) = 2x^2 - 3x - 4

Division statement: (4x^3 + 6x^2 - 6x - 9) = (2x + 3)(2x^2 - 3x - 4)

Answer 2

a) To divide f(x) = x^3 - 3x^2 + x + 2 by d(x) = x + 2, we can use long division:

_____________________
x + 2 | x^3 - 3x^2 + x + 2
-(x^3 + 2x^2)
_____________
-5x^2 + x
-(-5x^2 - 10x)
____________
11x + 2
- (11x + 22)
__________
-20

So, the quotient q(x) = x^2 - 5x + 11, and the remainder r(x) = -20.

The division statement is: f(x) = d(x)q(x) + r(x)
f(x) = (x + 2)(x^2 - 5x + 11) - 20
f(x) = x^3 - 3x^2 + x + 2

b) To divide f(x) = 2x^3 - 4x^2 - 3x + 5 by d(x) = x - 3, we can use long division:

_____________________
x - 3 | 2x^3 - 4x^2 - 3x + 5
-(2x^3 - 6x^2)
_____________
2x^2 - 3x + 5
-(2x^2 - 6x)
____________
3x + 5
- (3x - 9)
__________
14

So, the quotient q(x) = 2x^2 - 3x + 3, and the remainder r(x) = 14.

The division statement is: f(x) = d(x)q(x) + r(x)
f(x) = (x - 3)(2x^2 - 3x + 3) + 14
f(x) = 2x^3 - 4x^2 - 3x + 5

c) To divide f(x) = 3x^3 + 7x^2 + 5x + 1 by d(x) = 3x + 1, we can use long division:

_____________________
3x + 1 | 3x^3 + 7x^2 + 5x + 1
-(3x^3 + x^2)
_____________
6x^2 + 5x + 1
-(6x^2 + 2x)
____________
3x + 1
- (3x + 1)
__________
0

So, the quotient q(x) = 6x^2 + 3x + 1, and the remainder r(x) = 0.

The division statement is: f(x) = d(x)q(x) + r(x)
f(x) = (3x + 1)(6x^2 + 3x + 1) + 0
f(x) = 3x^3 + 7x^2 + 5x + 1

d) To divide f(x) = 4x^3 + 6x^2 - 6x - 9 by d(x) = 2x + 3, we can use long division:

_____________________
2x + 3 | 4x^3 + 6x^2 - 6x - 9
-(4x^3 + 6x^2)
_____________
-12x - 9
-(-12x - 18)
____________
9

So, the quotient q(x) = -2x - 6, and the remainder r(x) = 9.

The division statement is: f(x) = d(x)q(x) + r(x)
f(x) = (2x + 3)(-2x - 6) + 9
f(x) = 4x^3 + 6x^2 - 6x - 9

e) To divide f(x) = x^4 - x^3 + 2x^2 - 3x + 8 by d(x) = x - 4, we can use long division:

_____________________
x - 4 | x^4 - x^3 + 2x^2 - 3x + 8
-(x^4 - 4x^3)
_____________
3x^3 + 2x^2 - 3x + 8
-(3x^3 - 12x^2)
____________
14x^2 - 3x + 8
-(14x^2 - 56x)
__________
53x + 8
-(53x - 212)
__________
220

So, the quotient q(x) = (x^3 - 4x^2) + (3x^2 - 12x) + 14x + 53, and the remainder r(x) = 220.

The division statement is: f(x) = d(x)q(x) + r(x)
f(x) = (x - 4)((x^3 - 4x^2) + (3x^2 - 12x) + 14x + 53) + 220
f(x) = x^4 - x^3 + 2x^2 - 3x + 8

Answer 3

To divide polynomials, we use the long division method. Here's how you can solve each problem by long division:

a) Divide (x^3 - 3x^2 + x + 2) by (x + 2):

Step 1: Divide the first term of the dividend (x^3) by the first term of the divisor (x). This gives us x^2. Write x^2 above the line.

x^2
_______________
x + 2 | x^3 - 3x^2 + x + 2

Step 2: Multiply the divisor (x + 2) by x^2, which gives us x^3 + 2x^2. Write this product beneath the dividend and line them up:

x^3 + 2x^2
_______________
x + 2 | x^3 - 3x^2 + x + 2

Step 3: Subtract the product (x^3 + 2x^2) from the dividend (x^3 - 3x^2) and write the result below:

x^3 - 3x^2 + x + 2
- (x^3 + 2x^2)
_______________

After subtracting, we get -5x^2 + x.

Step 4: Bring down the next term (x) from the dividend:

x^3 - 3x^2 + x + 2
- (x^3 + 2x^2)
_______________
-5x^2 + x

Step 5: Repeat the process by dividing (-5x^2 + x) by (x + 2):

-5x
_______________
x + 2 | x^3 - 3x^2 + x + 2
- (x^3 + 2x^2)
_______________
-5x^2 + x

Dividing -5x^2 by x gives us -5x, which is written above the line.

Step 6: Multiply the divisor (x + 2) by -5x, which gives us -5x^2 - 10x. Write this product beneath the result and line them up:

-5x^2 - 10x
_______________
x + 2 | x^3 - 3x^2 + x + 2
- (x^3 + 2x^2)
_______________
-5x^2 + x

Step 7: Subtract the product (-5x^2 - 10x) from the current result (-5x^2 + x):

-5x^2 + x
- (-5x^2 - 10x)
_______________

After subtracting, we get 11x.

Step 8: Since there are no more terms to bring down from the dividend, the division is complete. The quotient is x^2 - 5x and the remainder is 11x. So we can write: f(x) = (x^2 - 5x) * (x + 2) + 11x.

This can also be expressed as: f(x) = (x + 2)(x^2 - 5x) + 11x.

b) Divide (2x^3 - 4x^2 - 3x + 5) by (x - 3):

For this problem and the remaining ones, you can follow the same steps of long division to obtain the quotient and remainder.