|
Unlimited Tutoring & Homework Help
|
Statement
If p(x), a polynomial in x is divided by x-a and the remainder = f (a) is zero, then (x-a) is a factor of p(x).Proof:
When p(x) is divided by x-a,
R = p(a) (by remainder theorem)p(x) = (x-a).q(x)+p(a)
(Dividend = Divisor x quotient + Remainder Division Algorithm)But p(a) = 0 is given.
Hence p(x) = (x-a).q(x)
p(x) = (x-a).q(x) + R
If (x-a) is a factor then the remainder should be zero (x-a divides p(x)exactly)
R=0By remainder theorem, R = p(a)
p(a)=0Illustrative Examples
Example 1:
Determine whether x-2 is a factor of x2-7x+10.
Suggested answer:
p(x) = x2-7x+10 is divided by x-2.
R = p(2)= 4-14+10=0, R=0

Example 2:
Determine whether x-3 is a factor of x3-3x2+4x-12.
Suggested answer:
p(x) = x3-3x2+4x-12 is divided by x-3.
R = p(3)= 33- 3 x 32 + 4 x 3-12
= 27-27+12-12=0
Example 3:
Show that x+1 is a factor of 2x3+5x2-9x-12.
Suggested answer:
Let p(x) = 2x3+5x2-9x-12
when p(x) is
by x+1 [x- (-1)],
R = p(-1)
= 2(-1)3+5(-1)2-9(-1)-12= -2+5+9-12
= -14+14 = 0R = 0

Example 4:
Find a so that x4+2x3-ax2+x-2 has (x+2) as its factor.
Suggested answer:
Let p(x) = x4+2x3-ax2+x-2
when p(x) is
by x+2 [x- (-2)],
R = p(-2)
= (-2)4+2(-2)3-a(-2)2-2-2= 16-16-4a-4
= -4a - 4As (x+2) is a factor, R=0
-4a - 4 = 0a = -1
Example 5:
If (x-2) and (x-3) are factors of x3+ax2+bx+12, find a and b.
Suggested answer:
Let p(x) = x3+ax2+bx+12
when p(x) is
x-2,
R = p(2)
= 23+a(2)2+b(2)+12= 8+4a+2b+12
= 20+4a+2bBut x-2 is a factor,
20+4a+2b = 0
when p(x) is
by x-3,
= 27+9a+3b+12
= 39+9a+3bBut x-3 is a factor,
39+9a+3b=0
[9a+3b=-39]
3
3a+b = -13 …(2)
Subtracting equation (1) from (2), we get,3a+b-2a-b = -13+10
a = -3Substitute a = -3 in equation (1),
2a+b = -10-6+b = -10
b = -4Example 6:
Factorise using factor theorem.
a) 2x2+x-3b) 5x2+6x-8
Suggested answer:
a) 2x2+x-3
Let p(x) = 2x2+x-3The factors of the constant 3 are +1, -1, +3, -3.
Let us find p(1) = 2+1-3 = 0

b) 5x2+6x-8
Suggested answer:
Let p(x) = 5x2+6x-8
Factors of the constant '8' are 1,-1,2,-2,4, -4 and 8,-8.Let us find the value of p(x) at each one of them to get one factor.
p(-2) = 20-12-8 = 0
Divide 5x2+6x-8 by x+2 to get the other factor.


