Problems 1-7 : Find the remainder using Remainder Theorem.
Problem 1 :
3x^{3 }+ 4x^{2 }- 5x + 8 is divided by x - 1
Problem 2 :
5x^{3 }+ 2x^{2 }- 6x + 12 is divided by x + 2
Problem 3 :
2x^{3 }- 4x^{2 }+ 7x + 6 is divided by x - 2
Problem 4 :
4x^{3 }- 3x^{2 }+ 2x - 4 is divided by x + 3
Problem 5 :
4x^{3} - 12x^{2} + 11x - 5 is divided by 2x - 1
Problem 6 :
8x^{4 }+ 12x^{3 }- 2x^{2 }- 18x + 14 is divided by x + 1
Problem 7 :
x^{3 }- ax^{2} - 5x + 2a is divided by x - a
Problem 8 :
If the polynomial (2x^{3 }- ax^{2 }+ 9x - 8) is divided by (x - 3), the remainder is 28. Find the value of a.
Problem 9 :
Find the value of m if (x^{3 }- 6x^{2 }+ mx + 60) leaves the remainder 2 when divided by (x + 2).
Problem 10 :
If (x - 1) divides (mx^{3 }- 2x^{2 }+ 25x - 26) without remainder find the value of m.
Problem 11 :
If the polynomials (x^{3 }+ 3x^{2 }- m) and (2x^{3 }- mx + 9) leave the same remainder when they are divided by (x - 2), find the value of m. Also find the remainder.
Problem 12 :
p(x) = (3x^{2} - 5)(x + k) - 20
In the polynomial p(x) defined above, k is a constant. If x divides p(x) without remainder, find the value of k.
1. Answer :
Let p(x) = 3x^{3 }+ 4x^{2 }- 5x + 8.
Equate the (x - 1) to zero and solve for x.
x - 1 = 0
x = 1
By Remainder Theorem, when p(x) is divided by x - 1, the remainder is p(1).
Remaider :
= p(1)
= 3(1)^{3 }+ 4(1)^{2 }- 5(1) + 8
= 3 + 4 - 5 + 8
= 10
2. Answer :
Let p(x) = 5x^{3 }+ 2x^{2 }- 6x + 12.
Equate the divisor (x + 2) to zero and solve for x.
x + 2 = 0
x = -2
By Remainder Theorem, when p(x) is divided by x + 2, the remainder is p(-2).
Remainder :
= p(-2)
= 5(-2)^{3 }+ 2(-2)^{2 }- 6(-2) + 12
= 5(-8) + 2(4) + 12 + 12
= 40 + 8 + 12 + 12
= -8
3. Answer :
Let p(x) = 2x^{3 }- 4x^{2 }+ 7x + 6.
Equate the divisor (x - 2) to zero and solve for x.
x - 2 = 0
x = 2
By Remainder Theorem, when p(x) is divided by x - 2, the remainder is p(2).
Remainder :
= p(2)
= 2(2)^{3 }- 4(2)^{2 }+ 7(2) + 6
= 2(8) - 4(4) + 14 + 6
= 16 - 16 + 14 + 6
= 20
4. Answer :
Let p(x) = 4x^{3 }- 3x^{2 }+ 2x - 4.
Equate the divisor (x + 3) to zero and solve for x.
x + 3 = 0
x = -3
By Remainder Theorem, when p(x) is divided by x + 3, the remainder is p(-3).
Remainder :
= p(-3)
= 4(-3)^{3 }- 3(-3)^{2 }+ 2(-3) - 4
= 4(-27) - 3(9) - 6 - 4
= -108 - 27 - 6 - 4
= -145
5. Answer :
Let p(x) = 4x^{3} - 12x^{2} + 11x - 5.
Equate the divisor (2x - 1) to zero and solve for x.
2x - 1 = 0
x = ½
By Remainder Theorem, when p(x) is divided by 2x - 1, the remainder is p(½).
Remainder :
= p(½)
= 4(½)^{3 }- 12(½)^{2} + 11(½) - 5
= 4(⅛) - 12(¼) + ¹¹⁄₂ - 5
= ½ - 3 + ¹¹⁄₂ - 5
= ½ + ¹¹⁄₂ -3 - 5
= 6 - 8
= -2
6. Answer :
Let p(x) = 8x^{4 }+ 12x^{3 }- 2x^{2 }- 18x + 14.
Equate the divisor (x + 1) to zero and solve for x.
x + 1 = 0
x = -1
By Remainder Theorem, when p(x) is divided by x + 1, the remainder is p(-1).
Remainder :
= p(-1)
= 8(-1)^{4 }+ 12(-1)^{3 }- 2(-1)^{2 }- 18(-1) + 14
= 8 - 12 - 2 + 18 + 14
= 40 - 14
= 26
7. Answer :
Let p(x) = x^{3 }- ax^{2} - 5x + 2a.
Equate the divisor (x - a) to zero and solve for x.
x - a = 0
x = a
By Remainder Theorem, when p(x) is divided by x - a, the remainder is p(a)..
Remainder :
= p(a)
= a^{3 }- a(a)^{2} - 5a + 2a
= a^{3 }- a^{3} - 3a
= -3a
8. Answer :
Let p(x) = 2x^{3 }- ax^{2 }+ 9x - 8.
Equate the divisor (x - 3) to zero and solve for x.
x - 3 = 0
x = 3
By Remainder Theorem, when p(x) is divided by x - 3, the remainder is p(3).
Given : When p(x) is divided by (x - 3), the remainder is 28.
p(3) = 28
2(3)^{3} - a(3)^{2} + 9(3) - 8 = 28
2(27) - a(9) + 27 - 8 = 28
54 - 9a + 19 = 28
73 - 9a = 28
73 - 28 = 9a
45 = 9a
5 = a
9. Answer :
Let p(x) = x^{3 }- 6x^{2 }+ mx + 60.
Equate the divisor (x + 2) to zero and solve for x.
x + 2 = 0
x = -2
By Remainder Theorem, when p(x) is divided by x + 2, the remainder is p(-2).
Given : When p(x) is divided by (x + 2), the remainder is 2.
(-2)^{3 }- 6(-2)^{2 }+ m(-2) + 60 = 2
-8 - 6(4) - 2m + 60 = 2
-8 - 24 - 2m + 60 = 2
28 - 2m = 2
28 - 2 = 2m
26 = 2m
13 = m
10. Answer :
Let p(x) = mx^{3 }- 2x^{2 }+ 25x - 26.
Equate the divisor (x - 1) to zero and solve for x.
x - 1 = 0
x = 1
By Remainder Theorem, when p(x) is divided by x - 1, the remainder is p(1).
Given : (x - 1) divides (mx^{3 }- 2x^{2 }+ 25x - 26) without remainder.
m(1)^{3 }- 2(1)^{2 }+ 25(1) - 26 = 0
m - 2 + 25 - 26 = 0
m - 3 = 0
m = 3
11. Answer :
Let p(x) = x^{3 }+ 3x^{2 }- m and q(x) = 2x^{3 }- mx + 9.
Equate the divisor (x - 2) to zero and solve for x.
x - 2 = 0
x = 2
When p(x) is divided by (x - 2) the remainder is p(2).
When q(x) is divided by (x - 2) the remainder is q(2).
Given : If p(x) and q(x) are divided by (x - 2), the remainder is same.
p(2) = q(2)
2^{3 }+ 3(2)^{2 }- m = 2(2)^{3 }- m(2) + 9
8 + 3(4)^{ }- m = 2(8)^{ }- 2m + 9
8 + 12^{ }- m = 16 - 2m + 9
20 - m = 25 - 2m
m = 5
12. Answer :
p(x) = (3x^{2} - 5)(x + k) - 20
Equating the divisor x to sero,
x = 0
When p(x) is divided by x, the remainder is p(0).
Given : x divides p(x) without reemainder.
p(0) = 0
(3(0)^{2} - 5)(0 + k) - 20 = 0
(0 - 5)k - 20 = 0
-5k - 20 = 0
-5k = 20
k = -4
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