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 (An *Bn * Cn )/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+nC1xn-1+nC2xn-2a2....+ nCn-1x1an-1+an
(x+a)n /x = [xn+nC1xn-1+nC2xn-2a2....+ nCn-1x1an-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
About Gourav Tomar
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