REMAINDER THEOREM WORKSHEET

Problems 1-7 : Find the remainder using Remainder Theorem.

Problem 1 :

3x³ + 4x² - 5x + 8 is divided by x - 1

Problem 2 :

5x³ + 2x² - 6x + 12 is divided by x + 2

Problem 3 :

2x³ - 4x² + 7x + 6 is divided by x - 2

Problem 4 :

4x³ - 3x² + 2x - 4 is divided by x + 3

Problem 5 :

4x³ - 12x² + 11x - 5 is divided by 2x - 1

Problem 6 :

8x⁴ + 12x³ - 2x² - 18x + 14 is divided by x + 1

Problem 7 :

x³ - ax² - 5x + 2a is divided by x - a

Problem 8 :

If the polynomial (2x³ - ax² + 9x - 8) is divided by (x - 3), the remainder is 28. Find the value of a.

Problem 9 :

Find the value of m if (x³ - 6x² + mx + 60) leaves the remainder 2 when divided by (x + 2).

Problem 10 :

If (x - 1) divides (mx³ - 2x² + 25x - 26) without remainder find the value of m.

Problem 11 :

If the polynomials (x³ + 3x² - m) and (2x³ - 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) = (3x² - 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.

Answers

1. Answer :

Let p(x) = 3x³ + 4x² - 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) = 5x³ + 2x² - 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) = 2x³ - 4x² + 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) = 4x³ - 3x² + 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) = 4x³ - 12x² + 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) =  8x⁴ + 12x³ - 2x² - 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) = x³ - ax² - 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)

= a³ - a(a)² - 5a + 2a

= a³ - a³ - 3a

= -3a

8. Answer :

Let p(x) = 2x³ - ax² + 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) = x³ - 6x² + 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) = mx³ - 2x² + 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 (mx³ - 2x² + 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) = x³ + 3x² - m and q(x) = 2x³ - 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)

2³ + 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) = (3x² - 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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