Factors of Polynomial.
1. If P(x) = x^3 + 5 x^2 + 5x + 1 Find following are the factor of polynomial or not ?
a). x + 1
b) x – 1
c). x – 1/3
d). x + ¼
e). x
f). 3x + 6
Solution.
a). P(x) = x^3 + 5 x^2 + 5x + 1
Put x + 1 = 0 => x = -1
P(-1) = (-1)^3 + 5(-1)^2 + 5(-1) + 1
= -1 + 5 – 5 + 1 = 0
The reminder is zero. So (x + 1) is a factor of this polynomial.
b). P(x) = x^3 + 5 x^2 + 5x + 1
Put x - 1 = 0 => x = 1
P(1) = (1)^3 + 5(1)^2 + 5(1) + 1
= 1 + 5 + 5 + 1 = 12
The reminder is not zero. So (x - 1) is not a factor of this polynomial.
c). P(x) = x^3 + 5 x^2 + 5x + 1
Put x – 1/3 = 0 => x = 1/3
P(1/3) = (1/3)^3 + 5(1/3)^2 + 5(1/3) + 1
= 1/27 + 5/9 + 5/3 + 1 = ( 1 + 15 + 45 + 27 ) / 27 = 88/27
The reminder is not zero. So (x - 1/3) is not a factor of this polynomial.
d). P(x) = x^3 + 5 x^2 + 5x + 1
Put x + 1/4 = 0 => x = -1/4
P(-1/4) = (-1/4)^3 + 5(-1/4)^2 + 5(-1/4) + 1
= -1/64 + 5/16 - 5/4 + 1 = ( -1 + 20 - 80 + 64 ) / 64 = 3/64
The reminder is not zero. So (x + 1/4) is not a factor of this polynomial.
e). P(x) = x^3 + 5 x^2 + 5x + 1
Put x = 0
P(0) = (0)^3 + 5(0)^2 + 5(0) + 1
= 0 + 0 + 0 + 1 = 1
The reminder is not zero. So (x) is not a factor of this polynomial.
f). P(x) = x^3 + 5 x^2 + 5x + 1
Put 3x + 6 = 0 => 3x = -6
X = -6/3 => x = -2
P(-2) = (-2)^3 + 5(-2)^2 + 5(-2) + 1
= -8 + 5 * 4 – 5 * 2 + 1 = -8 + 20 – 10 + 1 = 3
The reminder is not zero. So (3x + 6) is not a factor of this polynomial.
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| maths-ncert-solution |
Sol. P(x) = 3x^3 – a x^2 + 2 x – 2 a
Put x – a = 0 => x = a
P(a) = 3 a^3 – a * a^2 + 2 * a -2 a
= 3 a^3 – a^3 + 2a -2a
= 2 a^3
The reminder is not zero. So (x - a) is not a factor of this polynomial.
3. Find the reminder of P(x) = 27 x^3 – 9 x^2 + 3 x – 1 when divided by x – 1/3
Sol. P(x) = 27 x^3 – 9 x^2 + 3 x – 1
Put x – 1/3 = 0 => x = 1/3
P(1/3) = 27 * (1/3)^3 – 9 * (1/3)^2 + 3 * (1/3) - 1
= 1 – 1 + 1 – 1 = 0
The reminder is zero. So (x - 1/3) is a factor of this polynomial.
Find g(x) is a factor of P(x) or not
4. If P(x) = x^3 + 7 x^2 + 7 x + 1 and g(x) = x + 1Sol. g(x) = x + 1 is a factor of P(x) only when P(-1) must be zero.
P(x) = x^3 + 7 x^2 + 7 x + 1
Put x = -1
P(-1) = (-1)^3 + 7 * (-1)^2 + 7 * (-1) +1 = -1 + 7 - 7 + 1 = 0
So g(x) is a factor of P(x).
5. If P(x) = 9 x^2 + 3 x – 2 and g(x) = x - 1/3
Sol. g(x) = x - 1/3 is a factor of P(x) only when P(1/3) must be zero.
P(x) = 9 x^2 + 3 x - 2
Put x = 1/3
P(1/3) = 9 * (1/3)^2 + 3 * (1/3) - 2 = 1 + 1 - 2 = 0
So g(x) is a factor of P(x).
6. If P(x) = x^2 + x – 2 and g(x) = x + 1
Sol. g(x) = x + 1 is a factor of P(x) only when P(-1) must be zero.
P(x) = x^2 + x - 2
Put x = -1
P(-1) = (-1)^2 + (-1) - 2 = 1 - 1 - 2 = -2
So g(x) is not a factor of P(x).
7. If P(x) = 4 x^2 + 2 x – 2 and g(x) = x - 1/2
Sol. g(x) = x - 1/2 is a factor of P(x) only when P(1/2) must be zero.
P(x) = 4 x^2 + 2 x - 2
Put x = 1/2
P(1/2) = 4 * (1/2)^2 + 2 * (1/2) - 2 = 1 + 1 - 2 = 0
So g(x) is a factor of P(x).
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| maths-ncert-solution |
Sol. g(x) = x - 1 is a factor of P(x) only when P(1) must be zero.
P(x) = x^2 + x - 2
Put x = 1
P(1) = (1)^2 + (1) - 2 = 1 + 1 - 2 = 0
So g(x) is a factor of P(x).
9. If P(x) = 9 x^2 + 3 x – 2 and g(x) = x + 1/3
Sol. g(x) = x + 1/3 is a factor of P(x) only when P(1/3) must be zero.
P(x) = 9 x^2 + 3 x - 2
Put x = -1/3
P(-1/3) = 9 * (-1/3)^2 + 3 * (-1/3) - 2 = 1 - 1 - 2 = -2
So g(x) is not a factor of P(x).
10. If P(x) = x^3 + 7 x^2 + 7 x + 1 and g(x) = x - 1
Sol. g(x) = x - 1 is a factor of P(x) only when P(-1) must be zero.
P(x) = x^3 + 7 x^2 + 7 x + 1
Put x = 1
P(1) = (1)^3 + 7 * (1)^2 + 7 * (1) +1 = 1 + 7 + 7 + 1 = 16
So g(x) is not a factor of P(x).
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