Remainder theorem, polynomial theorem and factorization of polynomials

 

Remainder theorem:

Dividend = (divisor * quotient) + Remainder

Eg. If  26 is divided by 5 then  is 5 quotient, 5 is Divisor, 1 is Remainder, 26 is Divided

The remainder of expression (a*b*c)/n is equal to the remainder of expression (A_n *B_n * C_n )/n

Where An is remainder when a is divided by n

Bn  is the remainder when b is divided by n

Cn is the remainder when c is divided by n.

Polynomial theorem: This is a very powerful theorem to find the remainder.

According to the Polynomial theorem.

(x+a)ⁿ = xⁿ+nC₁x^n-1+nC₂x^n-2a²....+ nC_n-1x¹a^n-1+aⁿ

(x+a)ⁿ /x = [xⁿ+nC₁x^n-1+nC₂x^n-2a²....+ nC_n-1x¹a^n-1+aⁿ]/x

The remainder of expression will be equal to the remainder of aⁿ/x because the rest of the term contains x is completely divisible by x.

Ex. Find the remainder of (15*17*19)/7.

Sol. The remainder will be equal to (1*3*5)/7

15/7 = 1/7 =1 Ans

Ex. Find the remainder 9⁵⁰/7?

Sol. Using polynomial theorem.

 

(7+2)⁵⁰ =2⁵⁰/7 = [(2³)¹⁶ * 2²]/7

[(7+1)¹⁶ * 4]/7 = 1*4/7=4 Ans 

Remainder theorem for polynomials: This theorem represents the relationship between the divisor or the first degree in the form x-a and the remainder r(x).

Ex. without using the division process, find the remainder when x³+4x²+6x-2 is divided by x+5.

Sol. Step 1: Put divisor equal to 0. and find the value of x.

x+5 = 0

x = -5

Step 2. The remainder will be f(-5)

f(-5) = (-5)³ + 4(-5)²+6(-5)-2

= -125+100-30-2 = -57 Ans.

Ex. Find the value of P, if expression Px³+3x²-3 and 2x³-5x +P is divided by x-4 leave the same remainder.

Sol. The remainder is:

R1 = F(4)  = P(4)³ +3(4)²-3 = 64P+45

R2 = F(4) = 2(4)³ -5(4)+P = P+108

Since R1 = R2

64P+45 = P+108

63P = 63

P=1

Factorization of polynomials:

Factor Theorem: Let F(x) be a polynomial and a be a real number. Then two results hold.

i. if f(a)= 0 then x-a is a factor of f(x).

ii. if x-a if a factor of f(x) then f(a) = 0.

Ex. Let f(x) = x³ - 12x²+ 44x - 48 Find out whether x-2 and x-3 are factors of f(x).

Sol. a. x-2 = 0

x = 2

f(2) = 2³ - 12*2² + 44*2 -48 = 0 

Hence, x-2 is factor of f(x)

b. x-3= 0

x=3

f(3) = 3³ - 12*3²+44*3-48 = 3

Hence x-3 is not a factor of f(x).

Ex. Find whether 3x-1 is a factor of 27x³ - 9x² - 6x + 2 by the above rule.

Sol.we have

3x-1 = 0

x= 1/3

if 3x-1 is a factor of f(x) then f(1/3) should be equal to zero

f(1/3) = 27(1/3)³ - 9(1/3)² -6(1/3) + 2

 F(1/3) = 1-1-2+2 = 0

Conditions of divisibility:

1. xⁿ+aⁿ is exactly divisible by x+a only when n is odd.

Ex. a⁵+b⁵ is exactly divisible by a+b

2. xⁿ+aⁿ is not exactly divisible by x+a when n is even.

Ex. a⁸+b⁸ is not exactly divisible by a+b

3. xⁿ+aⁿ is never divisible by x-a.

Ex. a⁷-b⁷ or a¹⁰+b¹⁰ is not divisible by a-b.

4. xⁿ-aⁿ is exactly divisible by x+a.

Ex. x⁶-a⁶  is exactly divisible by x=a

5. xⁿ-aⁿ is exactly divisible by x-a.

Ex. x⁹-a⁹ and x¹⁰-a¹⁰ are exactly divisible by x-a