The remainder theorem states that when a polynomial p(x) is divided by (x-a), the remainder is equal to p(a). This is illustrated through several examples of polynomials being divided by expressions of the form (x-a) or (x+a), and calculating the remainder by evaluating p(a). The document also states that if p(a) equals 0 when a polynomial p(x) is divided by (x-a), then (x-a) is a factor of p(x). An example problem demonstrates finding the factor (x+1) of the polynomial p(x)=x^3-3x^2-9x-5 by using the remainder theorem.

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• If p(x) is divided
by (x-a) then
according to the
remainder theorem
,the remainder is
p(a).

4. EXAMPLES
• p(x)  x-a then the remainder will be p(a).
• p(x)  x-2 then the remainder will be p(2).
• p(x)  x-7 then the remainder will be p(7).
• p(x)  x-2/3 then the remainder will be p(2/3).
• p(x)  x+4 then the remainder will be p(-4).
• p(x)  x+3/7 then the remainder will be p(-3/7).
• p(x) 2x+3 then the remainder will be p(-3/2).
• p(x)  4+3x then the remainder will be p(-4/3).
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5. How to write the sums?
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Find the remainder theorem when x3+3x2+3x+1 is divided by x+1.
SOLUTION:
p(x)=x3+3x2+3x+1 ÷ x+1
According to the remainder theorem if we are going to divide
(x+1) then the remainder will be p(-1).
p(x) =x3+3x2+3x+1
p(-1)=(-1)3+3(-1)2+3(-1)+1
=(-1)+3(1)+3(-1)+1
=(-1)+3+(-3)+1
=0

6. If p(x) is a polynomial is divided
by (x-a):-
• If p(a)=0,then (x-a) is factor
of p(x).
• If (x-a) is a factor of p(x) then
p(a)=0.
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7. p(x)=x3-3x2-9x-5
g(x) =(x+1)
r(x)=(-1)[remainder]
p(x)=x3-3x2-9x-5
p(-1) = (-1)3-3(-1)2-9(-1)-5
=(-1)-3+9-5
=0
p(-1)=0(x+1) is the factor of p(x).
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EXAMPLE
• Find if g(x)=(x+1)is the factor of p(x)=x3-3x2-9x-5 or not ?

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