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Polynomials
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Question 1 of 10
1. Question
If the polynomials p(x) = 4x^{3}+3x^{2}+6 and f(x) = ax^{3}+ x^{2}+4 leave same remainder when divided by (x2) , then the value of a is
Correct
Explanation:
p(2) = 4 X 2^{3} + 3 X 2^{2} + 6 = 4 X 8 + 3 X 4 +6 = 32 +12 +6 = 46
f(x) = a X 2^{3} +2 X 2 + 4 = 8a+4+4 = 8a + 8
8a + 8 = 46
8a = 38
a = 38/8 = 19/4
Incorrect
Explanation:
p(2) = 4 X 2^{3} + 3 X 2^{2} + 6 = 4 X 8 + 3 X 4 +6 = 32 +12 +6 = 46
f(x) = a X 2^{3} +2 X 2 + 4 = 8a+4+4 = 8a + 8
8a + 8 = 46
8a = 38
a = 38/8 = 19/4

Question 2 of 10
2. Question
Making use of the remainder theorem, we find that the remainder of the polynomial p(x) = 5x^{3}+2x^{2}+1 when divided by (x2) is
Correct
Explanation:
x2 = 0 x=2
p(2) = 5 X 2^{3}+ 2X 2^{2}+1 = 5 X 8 + 2X 4 +1 = 40 + 8 + 1= 49
Incorrect
Correct Answer: Remainder = 49
Explanation:x2 = 0 x=2
p(2) = 5 X 2^{3}+ 2X 2^{2}+1 = 5 X 8 + 2X 4 +1 = 40 + 8 + 1= 49

Question 3 of 10
3. Question
The polynomial f(x) = 4x^{3}+ 4x^{2}+ px + q leaves the remainders as 3 and 4 when divided by (x2) and (x+3) .Find the values of p and q.
Correct
Correct Answer: p=31 , q=17
Explanation: p =31 , q = 17x – 2 =0 x = 2
f(2) = 4 X 2^{3} + 4 X 2^{2} +pX2 + q =3
32+16 + 2p + q = 3
2p + q = = 45 …. (i)
x+3 = 0 x =3
f (3) = 4 X (3)^{3} + 4 X (3)^{2 }+ (3) X p + q = 4
108 + 363p + q =4
72 3p + q = 4
3p + q = 76 …..(ii)
Solve equations (i) and (ii) to find values p and q.
Multiply (i) by 3 and (ii) by 2 we get,
6p + 3q = – 135
6p + 2q = 152
………………………. ( on Adding)
q = 17
Substituting the value of q = 17 in (i) we get
2p + q = 45
2p + 17 = 45
2p = 62
p = 31
Incorrect
Correct Answer: p =31 , q = 17
Explanation:
x – 2 =0 x = 2f(2) = 4 X 2^{3} + 4 X 2^{2} +pX2 + q =3
32+16 + 2p + q = 3
2p + q = = 45 …. (i)
x+3 = 0 x =3
f (3) = 4 X (3)^{3} + 4 X (3)^{2 }+ (3) X p + q = 4
108 + 363p + q =4
72 3p + q = 4
3p + q = 76 …..(ii)
Solve equations (i) and (ii) to find values p and q.
Multiply (i) by 3 and (ii) by 2 we get,
6p + 3q = – 135
6p + 2q = 152
………………………. ( on Adding)
q = 17
Substituting the value of q = 17 in (i) we get
2p + q = 45
2p + 17 = 45
2p = 62
p = 31

Question 4 of 10
4. Question
(x2) is a factor of p(x) = x^{3} – x^{2} + 3x 10 . Making use of factor theorem , state whether the given statement is true or false.
Correct
Explanation:
p(x) = x^{3} – x^{2} + 3x 10
p(2) = 2^{3 }2^{2} +3X2 10
= 8 – 4 + 6 10 = 0
Incorrect
Correct Answer: True
Explanation:p(x) = x^{3} – x^{2} + 3x 10
p(2) = 2^{3 }2^{2} +3X2 10
= 8 – 4 + 6 10 = 0

Question 5 of 10
5. Question
(x+3) is a factor of p(x) = x^{3 }+ x^{2} – 2x +12. Making use of factor theorem , state whether the given statement is true or false.
Correct
Explanation:
x+3 = 0 x =3
p(3) = (3)^{3 }+ (3)^{2 }– 2(3) +12 = 27 + 9 + 6 + 12 = 0
Incorrect
Correct Answer: True
Explanation:x+3 = 0 x =3
p(3) = (3)^{3 }+ (3)^{2 }– 2(3) +12 = 27 + 9 + 6 + 12 = 0

Question 6 of 10
6. Question
Find the value of b if (x+b) is the factor of 3x^{3}+3bx^{2}+4x+2
Correct
Explanation:
If (x+b) is the factor of p(x) = x^{3}+3x^{2}+4x+2 , then p(b) = 0
p(b) = 3(b)^{3} + 3b^{3} + 4b+ 2 = 0
3b^{3} + 3b^{3 }+ 4b +2 = 0
4b + 2 = 0
b = 2/4 = 1/2
Incorrect
Correct Answer : b = 1/2
Explanation:
If (x+b) is the factor of p(x) = x^{3}+3x^{2}+4x+2 , then p(b) = 0
p(b) = 3(b)^{3} + 3b^{3} + 4b+ 2 = 0
3b^{3} + 3b^{3 }+ 4b +2 = 0
4b + 2 = 0
b = 2/4 = 1/2

Question 7 of 10
7. Question
Find the value of a if (xa) is the factor of p(x) =x^{2}+ 5x+6
Correct
Explanation:
For (xa) to be factor of p(x), p(a) = 0 [ As xa = 0 x =a]
p(a) = a^{2}+ 5a + 6 = 0
a^{2} + 3a+2a + 6 = 0
a(a+3) + 2 (a+3) = 0
(a+3)(a+2) = 0
a=3 or a = 2
Incorrect
Correct Answer: a= – 3 or a = 2
Explanation:
For (xa) to be factor of p(x), p(a) = 0 [ As xa = 0 x =a]
p(a) = a^{2}+ 5a + 6 = 0
a^{2} + 3a+2a + 6 = 0
a(a+3) + 2 (a+3) = 0
(a+3)(a+2) = 0
a=3 or a = 2

Question 8 of 10
8. Question
For what value of a is p(x) = 4x^{2}+ax+6 exactly divisible by (x+3)
Correct
Explanation: For p(x) to be exactly divisible by (x+3) , p(3) must be equal to 0.
p(3) = 0
4 X (3)^{2 }+ a(3) + 6 = 0
36 3a + 6 = 0
3a = 30
a = 10
Incorrect
Correct Answer: a = 10
Explanation: For p(x) to be exactly divisible by (x+3) , p(3) must be equal to 0.p(3) = 0
4 X (3)^{2 }+ a(3) + 6 = 0
36 3a + 6 = 0
3a = 30
a = 10

Question 9 of 10
9. Question
For what value of b is p(x) = bx^{3 }+ 2x^{2}+3x+2 exactly divisible by (x2)
Correct
Explanation:
x2 = 0
x=2
For p(x) to be exactly divisible by (x2) , p(2) = 0
p(2) = 0
b x 23+ 2 X 22 + 3 X 2 + 2 = 0
8b + 8 + 6 + 2 = 0
8b + 16 = 0
b= 16/8 = 2Incorrect
Correct Answer: b = 2
Explanation:
x2=0
x=2
For p(x) to be exactly divisible by (x2) , p(2) = 0
p(2) = 0
b x 23+ 2 X 22 + 3 X 2 + 2 = 0
8b + 8 + 6 + 2 = 0
8b + 16 = 0
b= 16/8 = 2 
Question 10 of 10
10. Question
For what value of a is the polynomial f(x) = 2x^{3}+ax^{2}+5x+6 exactly divisible by (x+2)
Correct
Explanation: x + 2 = 0
x =2For f(x) to be exactly divisible by (x+2) , f(2) =0
f(2) = 2 X (2)^{3 }+ a x (2)^{2 }+ 5 (2) + 6 = 0
2 X (8) + a X (4) + (10) + 6 = 0
16 + 4a 10 +6 = 0
4a 20 = 0
a = 20/4 = 5
Incorrect
Correct Answer: a = 5
Explanation: x + 2 = 0
x =2For f(x) to be exactly divisible by (x+2) , f(2) =0
f(2) = 2 X (2)^{3 }+ a x (2)^{2 }+ 5 (2) + 6 = 0
2 X (8) + a X (4) + (10) + 6 = 0
16 + 4a 10 +6 = 0
4a 20 = 0
a = 20/4 = 5