Verify the division algorithm, i.e., Dividend = Divisor × Quotient + Remainder, in each of the following:I) 14x^2 + 13x - 15 divided by 7x - 4II) 15z^2 - 20z^2 - 13z - 12 divided by 3z - 6III) 34x - 22x^3 - 12x^4 - 10x^2 - 75 divided by 3x + 7IV) 15y^4 - 16y^3 - 9y^2 - (10/3)y + 6 divided by 3y - 2V) 6y^5 + 4y^4 + 4y^3 + 7y^2 + 27y + 6 divided by 2y^3 + 1VI) 4y^3 + 8y^2 + 8y + 7 divided by 2y^2 - y + 1

To verify the division algorithm for the given polynomials, we need to divide the dividend by the divisor, and then check if the division satisfies the formula:

Dividend = Divisor × Quotient + Remainder

We will perform polynomial division for each case and show the process step by step.

I) Divide 14x² + 13x - 15 by 7x - 4
Step 1: Perform the division of 14x² + 13x - 15 by 7x - 4.

Divide the first term of the dividend by the first term of the divisor:
14x² ÷ 7x = 2x

Multiply the divisor (7x - 4) by 2x:
2x(7x - 4) = 14x² - 8x

Subtract the result from the dividend:
(14x² + 13x - 15) - (14x² - 8x) = 21x - 15

Divide the first term of the new polynomial (21x) by the first term of the divisor (7x):
21x ÷ 7x = 3

Multiply the divisor by 3:
3(7x - 4) = 21x - 12

Subtract:
(21x - 15) - (21x - 12) = -3

So, the quotient is 2x + 3 and the remainder is -3.

Verification:
Dividend = Divisor × Quotient + Remainder
14x² + 13x - 15 = (7x - 4)(2x + 3) - 3
Expanding the right-hand side:
(7x - 4)(2x + 3) = 14x² + 21x - 8x - 12 = 14x² + 13x - 12
Thus, the equation becomes:
14x² + 13x - 15 = 14x² + 13x - 12 - 3, which is true.

II) Divide 15z² - 20z² - 13z - 12 by 3z - 6
First, simplify the dividend: 15z² - 20z² = -5z²

So, the dividend becomes:
-5z² - 13z - 12

Now, divide by 3z - 6:

Divide -5z² by 3z:
-5z² ÷ 3z = -5/3z

Multiply the divisor by -5/3z:
(-5/3z)(3z - 6) = -5z² + 10z

Subtract:
(-5z² - 13z - 12) - (-5z² + 10z) = -23z - 12

Divide -23z by 3z:
-23z ÷ 3z = -23/3

Multiply the divisor by -23/3:
(-23/3)(3z - 6) = -23z + 46

Subtract:
(-23z - 12) - (-23z + 46) = -58

So, the quotient is -5/3z - 23/3 and the remainder is -58.

Verification:
Dividend = Divisor × Quotient + Remainder
-5z² - 13z - 12 = (3z - 6)(-5/3z - 23/3) - 58

III) Divide 34x - 22x³ - 12x⁴ - 10x² - 75 by 3x + 7
Rearrange the dividend in standard form:
-12x⁴ - 22x³ - 10x² + 34x - 75

Now, divide by 3x + 7:

Divide -12x⁴ by 3x:
-12x⁴ ÷ 3x = -4x³

Multiply the divisor by -4x³:
(-4x³)(3x + 7) = -12x⁴ - 28x³

Subtract:
(-12x⁴ - 22x³ - 10x² + 34x - 75) - (-12x⁴ - 28x³) = 6x³ - 10x² + 34x - 75

Divide 6x³ by 3x:
6x³ ÷ 3x = 2x²

Multiply the divisor by 2x²:
(2x²)(3x + 7) = 6x³ + 14x²

Subtract:
(6x³ - 10x² + 34x - 75) - (6x³ + 14x²) = -24x² + 34x - 75

Divide -24x² by 3x:
-24x² ÷ 3x = -8x

Multiply the divisor by -8x:
(-8x)(3x + 7) = -24x² - 56x

Subtract:
(-24x² + 34x - 75) - (-24x² - 56x) = 90x - 75

Divide 90x by 3x:
90x ÷ 3x = 30

Multiply the divisor by 30:
(30)(3x + 7) = 90x + 210

Subtract:
(90x - 75) - (90x + 210) = -285

So, the quotient is -4x³ + 2x² - 8x + 30 and the remainder is -285.

IV) Divide 15y⁴ - 16y³ - 9y² - 10/3y + 6 by 3y - 2
Now divide by 3y - 2:

Divide 15y⁴ by 3y:
15y⁴ ÷ 3y = 5y³

Multiply the divisor by 5y³:
(5y³)(3y - 2) = 15y⁴ - 10y³

Subtract:
(15y⁴ - 16y³ - 9y² - 10/3y + 6) - (15y⁴ - 10y³) = -6y³ - 9y² - 10/3y + 6

Divide -6y³ by 3y:
-6y³ ÷ 3y = -2y²

Multiply the divisor by -2y²:
(-2y²)(3y - 2) = -6y³ + 4y²

Subtract:
(-6y³ - 9y² - 10/3y + 6) - (-6y³ + 4y²) = -13y² - 10/3y + 6

Divide -13y² by 3y:
-13y² ÷ 3y = -13/3y

Multiply the divisor by -13/3y:
(-13/3y)(3y - 2) = -13y² + 26/3y

Subtract:
(-13y² - 10/3y + 6) - (-13y² + 26/3y) = -36/3y + 6 = -12y + 6

So, the quotient is 5y³ - 2y² - 13/3y and the remainder is -12y + 6.

V) Divide 6y⁵ + 4y⁴ + 4y³ + 7y² + 27y + 6 by 2y³ + 1
Perform the division steps:

Divide 6y⁵ by 2y³:
6y⁵ ÷ 2y³ = 3y²

Multiply the divisor by 3y²:
(3y²)(2y³ + 1) = 6y⁵ + 3y²

Subtract:
(6y⁵ + 4y⁴ + 4y³ + 7y² + 27y + 6) - (6y⁵ + 3y²) = 4y⁴ + 4y³ + 4y² + 27y + 6

Divide 4y⁴ by 2y³:
4y⁴ ÷ 2y³ = 2y

Multiply the divisor by 2y:
(2y)(2y³ + 1) = 4y⁴ + 2y

Subtract:
(4y⁴ + 4y³ + 4y² + 27y + 6) - (4y⁴ + 2y) = 4y³ + 4y² + 25y + 6

Divide 4y³ by 2y³:
4y³ ÷ 2y³ = 2

Multiply the divisor by 2:
(2)(2y³ + 1) = 4y³ + 2

Subtract:
(4y³ + 4y² + 25y + 6) - (4y³ + 2) = 4y² + 25y + 4

So, the quotient is 3y² + 2y + 2 and the remainder is 4y² + 25y + 4.

VI) Divide 4y³ + 8y² + 8y + 7 by 2y² - y + 1
Perform the division:

Divide 4y³ by 2y²:
4y³ ÷ 2y² = 2y

Multiply the divisor by 2y:
(2y)(2y² - y + 1) = 4y³ - 2y² + 2y

Subtract:
(4y³ + 8y² + 8y + 7) - (4y³ - 2y² + 2y) = 10y² + 6y + 7

Divide 10y² by 2y²:
10y² ÷ 2y² = 5

Multiply the divisor by 5:
(5)(2y² - y + 1) = 10y² - 5y + 5

Subtract:
(10y² + 6y + 7) - (10y² - 5y + 5) = 11y + 2

So, the quotient is 2y + 5 and the remainder is 11y + 2.

These are the steps and verifications for all the divisions, confirming that the division algorithm holds in each case.