In the division by synthetic method the divisor polynomial is a whose degree is 1. in the form x + a or X Practice set 3.3 1. Divide each of the following polynomials by synthetic division method and linear division method. Write the quotient and the remainder. (i) (2m2 - 3m + 10) = (m - 5) (ii) (x4 + 2x3 + 3x2 + 4x + 5) = (x + 2) + (iii) (13 - 216 ; ( 22 23 1 17
Answers
Step-by-step explanation:
. Synthetic division:
(2m2 – 3m + 10) ÷ (m – 5)
Dividend = 2m2 – 3m + 10
∴ Coefficient form of dividend = (2, -3, 10)
Divisor = m – 5
∴ Opposite of -5 is 5.
Coefficient form of quotient = (2, 7)
∴ Quotient = 2m + 7,
Remainder = 45
Linear division method:
2m2 – 3m + 10 To get the term 2m2,
multiply (m – 5) by 2m and
add 10m, = 2m(m – 5) + 10m- 3m + 10 = 2m(m –
5) + 7m + 10 To get the term 7m,
multiply (m – 5) by 7
and add 35 = 2m(m – 5) + 7(m- 5) + 35+ 10
= (m – 5) (2m + 7) + 45
∴ Quotient = 2m + 7,
Remainder = 45
ii. Synthetic division:
(x4 + 2x3 + 3x2 + 4x + 5) ÷ (x + 2)
Dividend = x4 + 2x3 + 3x2 + 4x + 5
∴ Coefficient form of dividend = (1, 2, 3, 4, 5)
Divisor = x + 2
∴ Opposite of + 2 is -2
Coefficient form of quotient = (1, 0, 3, -2)
∴ Quotient = x3 + 3x – 2,
Remainder = 9
Linear division method:
x4 + 2x3 + 3x2 + 4x + 5
To get the term x4,
multiply (x + 2) by x3
and subtract 2x3,
= x3(x + 2) – 2x3 + 2x3 + 3x2 + 4x + 5
= x3(x + 2) + 3x2 + 4x + 5
To get the term 3x2,
multiply (x + 2) by 3x and subtract 6x,
= x3(x + 2) + 3x(x + 2) – 6x + 4x + 5
= x3(x + 2) + 3x(x + 2) – 2x + 5
To get the term -2x,
multiply (x + 2) by -2
and add 4,
= x3(x + 2) + 3x(x + 2) – 2(x + 2) + 4 + 5
= (x + 2) (x3 + 3x – 2) + 9
∴ Quotient = x3 + 3x – 2,
Remainder – 9
iii. Synthetic division:
(y3 – 216) ÷ (y – 6)
Dividend = y3 – 216
∴ Index form = y3 + 0y3 + 0y – 216
∴ Coefficient form of dividend = (1, 0, 0, -216)
Divisor = y – 6
∴ Opposite of – 6 is 6.
Coefficient form of quotient = (1, 6, 36)
∴ Quotient = y2 + 6y + 36,
Remainder = 0
Linear division method:
y3 – 216
To get the term y3,
multiply (y – 6) by y2
and add 6y2, = y2(y – 6) + 6y2 – 216
= y2(y – 6) + 6ysup>2 – 216
To get the, term 6y2
multiply (y – 6) by 6y
and add 36y,
= y2(y – 6) + 6y(y – 6) + 36y – 216 = y2(y – 6)
+ 6y(y – 6) + 36y – 216
To get the term 36y, multiply (y- 6) by 36
and add 216, = y2(y – 6) + 6y(y – 6) + 36(y – 6)
+ 216 – 216 = (y – 6) (y2 + 6y + 36) + 0
Quotient = y2 + 6y + 36
Remainder = 0