REMAINDER THEOREM AND FACTOR THEOREM
About "Remainder theorem and Factor theorem"
Remainder theorem and Factor theorem :
In this section, we are going to see remainder and factor theorem.
Remainder theorem :
If a polynomial P(x) is divided by (x-a), the remainder is P(a).
Using division algorithm, we have
P(x) = Q(x)(x-a) + P(a)
Here, Q(x) is the quotient when P(x) is divided by (x-a).
Factor theorem :
A polynomial P(x) would have a factor (x-a), if and only if P(a) = 0.
Remainder theorem and Factor theorem - Examples
Example 1 :
Using Remainder theorem, find the remainder when the polynomial 3x³- 2x² + 6x - 7 is divided by (x-2).
Solution :
Step 1 :
Let P(x) = 3x³- 2x² + 6x - 7.
If x - 2 = 0, then x = 2.
Step 2 :
To know the remainder when P(x) divided by (x-2), aaaaa plug x = 2 in P(x).
Remainder = P(2)
Remainder = 3(2)³- 2(2)² + 6(2) - 7
Remainder = 24 - 8 + 12 - 7
Remainder = 21
Example 2 :
Using Remainder theorem, find the remainder when the polynomial 7x⁴ - x²- 3x + 9 is divided by (x-6).
Solution :
Step 1 :
Let P(x) = 7x⁴ - x²- 3x + 9
If x - 6 = 0, then x = 6.
Step 2 :
To know the remainder when P(x) divided by (x-6), aaaaa plug x = 6 in P(x).
Remainder = P(6)
Remainder = 7(6)⁴ - (6)² - 3(6) + 9
Remainder = 9072 - 36 - 18 + 9
Remainder = 9027
Example 3 :
Using Remainder theorem, find the remainder when the polynomial x³+ 3x² - 5x + 2 is divided by (x+5).
Solution :
Step 1 :
Let P(x) = x³+ 3x² - 5x + 2
If x + 5 = 0, then x = - 5.
Step 2 :
To know the remainder when P(x) divided by (x+5), aaaaa plug x = - 5 in P(x).
Remainder = P(-5)
Remainder = (-5)³+ 3(-5)² - 5(-5) + 2
Remainder = -125 + 3(25) + 25 + 2
Remainder = -125 + 75 + 25 + 2
Remainder = -23
Example 4 :
Using Factor theorem, check whether (x-2) is a factor of the polynomial x⁴ - 3x³ + 2x² + 8x - 16.
Solution :
Step 1 :
Let P(x) = x⁴ - 3x³ + 2x² + 8x - 16
If x - 2 = 0, then x = 2.
Step 2 :
Using Factor theorem, to check whether (x-2) is factor of P(x), plug x = 2 in P(x).
P(2) = (2)⁴ - 3(2)³ + 2(2)² + 8(2) - 16
P(2) = 16 - 24 + 8 + 16 - 16
P(2) = 0
Because P(2) = 0, by Factor theorem, (x-2) is a factor of the polynomial P(x).
Example 5 :
Using Factor theorem, check whether (x+4) is a factor of the polynomial x² - 8x + 16.
Solution :
Step 1 :
Let P(x) = x² - 8x + 16
If x + 4 = 0, then x = -4.
Step 2 :
Using Factor theorem, to check whether (x+4) is factor of P(x), plug x = -4 in P(x).
P(-4) = (-4)² + 8(-4) + 16
P(-4) = 16 - 32 + 16
P(-4) = 0
Because P(-4) = 0, by Factor theorem, (x+4) is a factor of the polynomial P(x).
Example 6 :
Using Factor theorem, check whether (x-3) is a factor of the polynomial x³ - 2x² + 5x + 6.
Solution :
Step 1 :
Let P(x) = x³ - 2x² + 5x + 6
If x - 3 = 0, then x = 3.
Step 2 :
Using Factor theorem, to check whether (x-3) is factor of P(x), plug x = 3 in P(x).
P(3) = (3)³ - 2(3)² + 5(3) + 6
P(3) = 27 - 2(9) + 15 + 6
P(3) = 27 - 18 + 15 + 6
P(3) = 30 ≠ 0
Because P(3) ≠ 0, by Factor theorem, (x-3) is not a factor of the polynomial P(x).
After having gone through the stuff given above, we hope that the students would have understood "Remainder theorem and Factor theorem".
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