(i)
Dividend = Divisor Quotient + Remainder
14x2 + 13x - 15 = (7x - 4) x (2x + 3) + (-3)
14x2 + 13x - 15 = 14x2 + 21x - 8x - 12 - 3
14x2 + 13x - 15 = 14x2 + 13x - 15
(ii)
Dividend = Divisor Quotient + Remainder
15z3 - 20z + 13z - 12 = (3z - 6) x (5z2 + \(\frac{10z}{3}\) + 11) + 54
15z3 - 20z + 13z - 12 = 15z3 + 10z2 + 33z - 30z2 - 20z + 54
15z3 - 20z + 13z - 12 = 15z3 - 20z2 + 13z - 12
(iii)
Dividend = Divisor Quotient + Remainder
6y5 - 28y3 + 3y2 + 30y - 9 = (2y2 - 6) x (3y3 - 5y + \(\frac{3}{2}\)) + 0
6y5 - 28y3 + 3y2 + 30y - 9 = 6y5 - 10y3 + 3y2 - 18y3 + 30y - 9
6y5 - 28y3 + 3y2 + 30y - 9 = 6y5 - 28y3 + 3y2 + 30y - 9
(iv)
Dividend = Divisor Quotient + Remainder
-12x4 - 22x - 10x2 + 34x - 75 = (3x + 7) x (-4x3 + 2x2 - 8x + 30) - 285
-12x4 - 22x - 10x2 + 34x - 75 = -12x4 + 6x3 - 24x2 - 28x3 + 14x2 + 90x - 56x + 210 - 285
-12x4 - 22x - 10x2 + 34x - 75 = -12x4 - 22x3 - 10x3 + 34x - 75
(v)
Dividend = Divisor Quotient + Remainder
15y4 - 16y3 + 9y2 - \(\frac{10y}{3}\) + 6 = (3y - 2) x (5y3 - 2y + \(\frac{5y}{3}\)) + 6
15y4 - 16y3 + 9y2 - \(\frac{10y}{3}\) + 6 = 15y4 - 6y3 + 5y2 - 10y3 + 4y2 - \(\frac{10y}{3}\) + 6
15y4 - 16y3 + 9y2 - \(\frac{10y}{3}\) + 6 = 15y4 - 16y3 + 9y2 - \(\frac{10y}{3}\) + 6
(vi)
Dividend = Divisor Quotient + Remainder
4y3 + 8y2 + 84y + 7 = (2y2 - y + 1) x (2y + 5) + 11y + 2
4y3 + 8y2 + 84y + 7 = 4y3 + 10y2 - 2y2 -5y + 2y +5 + 11y + 2
4y3 + 8y2 + 84y + 7 = 4y3 + 8y2 + 8y + 7
(vii)
Dividend = Divisor Quotient + Remainder
6y5 - 4y4 + 4y3 + 7y2 + 27y + 6 = (2y3 + 1) x (3y2 + 2y + 2) + 4y2 + 25y + 4
6y5 - 4y4 + 4y3 + 7y2 + 27y + 6 = 6y5 + 4y4 + 4y3 + 3y2 + 2y + 2 + 4y2 + 25y + 4
6y5 - 4y4 + 4y3 + 7y2 + 27y + 6 = 6y5 + 4y4 + 4y3 + 7y2 + 27y + 6