Solve all the given questions with steps. Wrong or incorrect answers will be reported
Answers
[1]
The remainder theorem states that a polynomial p(x) will leave same remainder when divided by g(x) and the factors of g(x)
So, We have,
- p(x) = x³ - 4x² - 3x + 10
- g(x) = x - 4
⇒ g(x) = 0 [ to find the factors ]
⇒ x - 4 = 0
⇒ x = 4
Here, The factor of g(x) is 4
So, The remainder when p(x) is divided by g(x), we have
⇒ (4)³ - 4(4)² - 3(4) + 10
⇒ 64 - 64 - 12 + 10
⇒ 10 - 12
⇒ -2
Hence, remainder will be -2.
[2]
Same as the question 1.
- p(x) = x³ - 6x² + 2x - 4
- g(x) = 1 - 2x
To find the factor of g(x), evaluate it to zero because a factor of a polynomial satisfies the polynomial and evaluates it to zero.
⇒ g(x) = 0
⇒ 1 - 2x = 0
⇒ -2x = -1
⇒ x = 1/2
Put, x = 1/2 in p(x) to find the remainder when it is divided by g(x),
⇒ (1/2)³ - 6(1/2)² + 2(1/2) - 4
⇒ 1/8 - 3/2 + 1 - 4
⇒ (1 - 12)/8 - 3
⇒ (-11 - 24)/8
⇒ -35/8
Hence, The remainder is -35/8
[3]
We are given,
- p(x) = 2x³ - 2x² - 2x - 5
- g(x) = x - 1
if we substitute x in p(x) with any factor of g(x) we would get the same remainder when p(x) is divided by g(x)
⇒ g(x) = 0
⇒ x - 1 = 0
⇒ x = 0
Now, Substitute x = 1 in p(x), we have
⇒ 2(1)³ - 2(1)² - 2(1) - 5
⇒ 2 - 2 - 2 - 5
⇒ -7
Hence, The remainder is -7.
∴ Option D is correct.
[4]
Given :-
- f(x) = x³ - 6x² + 2x - 4
- g(x) = 3x - 1
To Find :-
- Remainder when p(x) is divided by g(x)
Solution :-
Same as previous problems, The value we would get after substituting x in p(x) with any factor of g(x) is the same that we would get after dividing p(x) by g(x),
Finding factor of g(x)
⇒ g(x) = 0
⇒ 3x - 1 = 0
⇒ 3x = 1
⇒ x = 1/3
Now, Substituting x = 1/3 in p(x)
⇒ (1/3)³ - 6(1/3)² + 2(1/3) - 4
⇒ 1/27 - 2/3 + 2/3 - 4
⇒ 1/27 - 4
⇒ (1 - 108)/4
⇒ -107/4
For the verification, refer to the attachment provided.
[5]
Given :-
- p(x) = ax³ + 3x² - 3
- g(x) = 2x³ - 5x + a
- f(x) = x - 4
It is given that, p(x) when divided by f(x) leaves the same remainder when f(x) divides g(x),
According to the remainder theorem, When substituting x in p(x) and g(x) with any factor of f(x), It would still leave the same remainder.
Finding factor of f(x),
⇒ f(x) = 0
⇒ x - 4 = 0
⇒ x = 4
Now, that we have got a factor of f(x) i.e., 4 so as given in the question,
⇒ p(4) = g(4)
⇒ a(4)³ + 3(4)² - 3 = 2(4)³ - 5(4) + a
⇒ 64a + 48 - 3 = 128 - 20 + a
⇒ 63a = 108 - 45
⇒ 63a = 63
⇒ a = 1
Hence, The value of a is 1.
[6]
Given :-
- f(x) = x³ - 6x² + 11x - 6
- g(x) = x - 3
A factor of a polynomial satisfies the polynomial in other words it makes the polynomial produce 0 when substituted in the place of x.
So, If g(x) is a factor of p(x) then the factor of g(x) will also satisfy p(x).
Finding factor of g(x),
⇒ g(x) = 0
⇒ x - 3 = 0
⇒ x = 3
Now, substituting x = 3 in p(x)
⇒ (3)³ - 6(3)² + 11(3) - 6
⇒ 27 - 54 + 33 - 6
⇒ 60 - 60
⇒ 0
Since, 4 satisfies p(x) and is also a factor of g(x) then according to the factor theorem, g(x) is a factor of p(x).
[7]
Same as (6)
[8]
To find whether the x - 1 is a factor of the given polynomials, substitute x = 1 in the given polynomials and the polynomials that have x - 1 as factor will evaluate to zero.
(a) x¹² - 1
⇒ (1)¹² - 1
⇒ 1 - 1
⇒ 0
(b) x²⁰²⁰ - 1
⇒ (1)²⁰²⁰ - 1
⇒ 1 - 1
⇒ 0
(c) 4x³ + 3x² + 5x - 12
⇒ 4(1)³ + 3(1)² + 5(1) - 12
⇒ 4 + 3 + 5 - 12
⇒ 12 - 12
⇒ 0
(d) all
Since, every polynomial that is given in the options has the common factor x - 1
Hence, Option (d) is correct.
[9]
p(x) = x³ - 1
To find the factor, evaluate it to zero
⇒ x³ - 1 = 0
⇒ x³ = 1
⇒ x = 1
⇒ x - 1 = 0
Hence, One of the factors of p(x) is x - 1
∴ Option (C) is correct.
[10]
- p(x) = x³ + ax² - 2x + a + 4
- g(x) = x + a
It is given that g(x) is a factor of p(x) hence, any factor of g(x) would also be a factor of p(x) in other words the remainder when p(x) is divided by g(x) is 0.
⇒ g(x) = 0
⇒ x + a = 0
⇒ x = -a
Substituting x = -a in p(x), we have
⇒ (-a)³ + a(a)² - 2(a) + a + 4 = 0
⇒ -a³ + a³ - 2a + a = - 4
⇒ -a = -4
⇒ a = 4
Hence, The value of a is 4.
