This set of Numerical Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Synthetic Division of a Polynomial by a Linear Expression”.

1. What will be the quotient by synthetic division on the following: (4x^{3} – 3x^{2} + 2x – 4) / (x + 1)?

a) 4x^{2} – 7x + 9

b) 4x^{2} + 7x – 9

c) 4x^{2} – 5x + 3

d) 4x^{2} + 5x – 3

View Answer

Explanation: Given,

To determine root divisor, we have to solve divisor equation x + 1 = 0

Therefore, our root becomes x = – 1

Write coefficients of the dividend 4x

^{3}– 3x

^{2}+ 2x – 4 to the right and our root -1 to the left.

-1 | 4 | -3 | 2 | -4 |

Step-1: Write down the first coefficient 4

-1 | 4 | -3 | 2 | -4 |

4 |

Step-2: Multiply our root -1 by our last result 4 to get -4 [(-1) × 4 = – 4]

-1 | 4 | -3 | 2 | -4 |

-4 | ||||

4 |

Step-3: Add new result -4 to the next coefficient of the dividend -3, and write down the sum -7, [(-3) + (-4) =-7]

-1 | 4 | -3 | 2 | -4 |

-4 | ||||

4 | -7 |

Step-4: Multiply our root -1 by our last result -7 to get 7 [(-1) × (-7) =7]

-1 | 4 | -3 | 2 | -4 |

-4 | 7 | |||

4 | -7 |

Step-5: Add new result 7 to the next coefficient of the dividend 2, and write down the sum 9, [2 + 7=9]

-1 | 4 | -3 | 2 | -4 |

-4 | 7 | |||

4 | -7 | 9 |

Step-6: Multiply our root -1 by our last result 9 to get -9 [(-1) × 9=-9]

-1 | 4 | -3 | 2 | -4 |

-4 | 7 | -9 | ||

4 | -7 | 9 |

Step-7: Add new result -9 to the next coefficient of the dividend -4, and write down the sum -13, [(-4) +(-9) =-13]

-1 | 4 | -3 | 2 | -4 |

-4 | 7 | -9 | ||

4 | -7 | 9 | -13 |

We have completed the table and have obtained the following coefficients

4, -7, 9, -13

All coefficients, except last one, are coefficients of quotient, last coefficient is remainder.

Thus, quotient is 4x^{2} – 7x + 9 and remainder is – 13.

2. What will be the remainder by synthetic division on the following: (2x^{4} + 11x^{3} + 13x^{2} + 2x – 8) / (x + 4)?

a) 4

b) 7

c) 3

d) 0

View Answer

Explanation: Given,

To determine root divisor, we have to solve divisor equation x + 4 = 0

Therefore, our root becomes x = -4.

Write coefficients of the dividend 2x

^{4}+ 11x

^{3}+ 13x

^{2}+ 2x – 8 to the right and our root -4 to the left.

-4 | 2 | 11 | 13 | 2 | -8 |

Step-1: Write down the first coefficient 2

-4 | 2 | 11 | 13 | 2 | -8 |

2 |

Step-2: Multiply our root -4 by our last result 2 to get -8 [(-4) × 2=-8]

-4 | 2 | 11 | 13 | 2 | -8 |

-8 | |||||

2 |

Step-3: Add new result -8 to the next coefficient of the dividend 11, and write down the sum 3, [11 + (-8) = 3]

-4 | 2 | 11 | 13 | 2 | -8 |

-8 | |||||

2 | 3 |

Step-4: Multiply our root -4 by our last result 3 to get -12 [(-4) × 3=-12]

-4 | 2 | 11 | 13 | 2 | -8 |

-8 | -12 | ||||

2 | 3 |

Step-5: Add new result -12 to the next coefficient of the dividend 13, and write down the sum 1, [13 + (-12) =1]

-4 | 2 | 11 | 13 | 2 | -8 |

-8 | -12 | ||||

2 | 3 | 1 |

Step-6: Multiply our root -4 by our last result 1 to get -4 [(-4) × 1=-4]

-4 | 2 | 11 | 13 | 2 | -8 |

-8 | -12 | -4 | |||

2 | 3 | 1 |

Step-7: Add new result -4 to the next coefficient of the dividend 2, and write down the sum -2, [2 + (-4) =-2]

-4 | 2 | 11 | 13 | 2 | -8 |

-8 | -12 | -4 | |||

2 | 3 | 1 | -2 |

Step-8: Multiply our root -4 by our last result -2 to get 8 [(-4) × (-2) =8]

-4 | 2 | 11 | 13 | 2 | -8 |

-8 | -12 | -4 | 8 | ||

2 | 3 | 1 | -2 |

Step-9: Add new result 8 to the next coefficient of the dividend -8, and write down the sum 0, [(-8) +

8=0]

-4 | 2 | 11 | 13 | 2 | -8 |

-8 | -12 | -4 | 8 | ||

2 | 3 | 1 | -2 | 0 |

We have completed the table and have obtained the following coefficients

2, 3, 1,-2, 0

All coefficients, except last one, are coefficients of quotient, last coefficient is remainder.

Thus, quotient is 2x^{3} + 3x^{2} + x – 2 and remainder is 0.

3. Synthetic division is generally used for dividing out factors.

a) True

b) False

View Answer

Explanation: Synthetic division is not used for the purpose of dividing out factors. Synthetic division is generally used for finding roots of polynomials and not for the purpose of dividing out factors as it is a shorthand method for polynomial division.

4. We can find the originating polynomial by going backwards from the roots.

a) False

b) True

View Answer

Explanation: For occurrence, if we are specified that x = –1 and x = –3 are the roots of a quadratic, then we know that x + 1 = 0, so x + 1 is a factor, and x + 3 = 0, so x + 3 is a factor. Consequently, we know that the quadratic must be of the form y = a(x + 3) (x + 1).

5. What should be the leading coefficient of linear expression, while dividing a polynomial by synthetic division?

a) 0

b) 1

c) 2

d) 3

View Answer

Explanation: In demand to divide polynomials using synthetic division, we must be dividing by a linear expression and the foremost coefficient (initial number) must be a 1. If the foremost coefficient is not 1, then we must divide by the foremost coefficient to turn the foremost coefficient into 1.

6. What will be the quotient by synthetic division on the following: (- 11x^{4} + 2x^{3} – 8x^{2} – 4) / (x + 1)?

a) -11x^{3} + 13x^{2} – 21x + 21

b) -11x^{3} + 13x^{2} – 21x – 21

c) -11x^{3} + 13x^{2} – 21

d) -11x^{3} + 13x^{2} + 21

View Answer

Explanation: Given,

To determine root divisor, we have to solve divisor equation x+1=0

Therefore, our root becomes x=-1

Write coefficients of the dividend – 11x^{4} + 2x^{3} – 8x^{2} – 4 to the right and our root -1 to the left

-1 | -11 | 2 | -8 | 0 | -4 |

Step-1: Write down the first coefficient -11

-1 | -11 | 2 | -8 | 0 | -4 |

-11 |

Step-2: Multiply our root -1 by our last result -11 to get 11 [(-1) × (-11) = 11].

-1 | -11 | 2 | -8 | 0 | -4 |

11 | |||||

-11 |

Step-3: Add new result 11 to the next coefficient of the dividend 2, and write down the sum 13, [2+11=13]

-1 | -11 | 2 | -8 | 0 | -4 |

11 | |||||

-11 | 13 |

Step-4: Multiply our root -1 by our last result 13 to get -13 [(-1) × 13=-13]

-1 | -11 | 2 | -8 | 0 | -4 |

11 | -13 | ||||

-11 | 13 |

Step-5: Add new result -13 to the next coefficient of the dividend -8, and write down the sum -21, [(-8) +(-13) =-21].

-1 | -11 | 2 | -8 | 0 | -4 |

11 | -13 | ||||

-11 | 13 | -21 |

Step-6: Multiply our root -1 by our last result -21 to get 21 [(-1) × (-21) =21]

-1 | -11 | 2 | -8 | 0 | -4 |

11 | -13 | 21 | |||

-11 | 13 | -21 |

Step-7: Add new result 21 to the next coefficient of the dividend 0, and write down the sum 21, [0+21=21]

-1 | -11 | 2 | -8 | 0 | -4 |

11 | -13 | 21 | |||

-11 | 13 | -21 | 21 |

Step-8: Multiply our root -1 by our last result 21 to get -21 [(-1) × 21=-21]

-1 | -11 | 2 | -8 | 0 | -4 |

11 | -13 | 21 | -21 | ||

-11 | 13 | -21 | 21 |

Step-9: Add new result -21 to the next coefficient of the dividend -4, and write down the sum -25, [(-4) + (-21) =-25]

-1 | -11 | 2 | -8 | 0 | -4 |

11 | -13 | 21 | -21 | ||

-11 | 13 | -21 | 21 | -25 |

We have completed the table and have obtained the following coefficients

-11, 13,-21, 21,-25

All coefficients, except last one, are coefficients of quotient, last coefficient is remainder.

Thus, quotient is -11x^{3} + 13x^{2} – 21x + 21 and remainder is -25.

7. What will be the remainder by synthetic division on the following: (4x^{2} – 24x + 35) / (x – 5)?

a) 15

b) 25

c) 05

d) 35

View Answer

Explanation: Given,

To determine root divisor, we have to solve divisor equation x-5=0

Therefore, our root becomes x=5

Write coefficients of the dividend 4x

^{2}– 24x + 35 to the right and our root 5 to the left.

5 | 4 | -24 | 35 |

Step-1: Write down the first coefficient 4

5 | 4 | -24 | 35 |

4 |

Step-2: Multiply our root 5 by our last result 4 to get 20 [5 × 4=20]

5 | 4 | -24 | 35 |

20 | |||

4 |

Step-3: Add new result 20 to the next coefficient of the dividend -24, and write down the sum -4, [(-24) + 20 =-4]

5 | 4 | -24 | 35 |

20 | |||

4 | -4 |

Step-4: Multiply our root 5 by our last result -4 to get -20 [5 × (-4) =-20]

5 | 4 | -24 | 35 |

20 | -20 | ||

4 | -4 |

Step-5: Add new result -20 to the next coefficient of the dividend 35, and write down the sum 15, [35+(-20) =15]

5 | 4 | -24 | 35 |

20 | -20 | ||

4 | -4 | 15 |

We have completed the table and have obtained the following coefficients

4,-4, 15

All coefficients, except last one, are coefficients of quotient, last coefficient is remainder.

Thus, quotient is 4x-4 and remainder is 15.

8. Choose the incorrect option regarding synthetic division.

a) It necessitates less steps for solution

b) The calculation can be completed without demanding the variables while solving

c) It is a less miscalculation prone process

d) It is a longhand method

View Answer

Explanation: Synthetic division is a short hand method as it requires fewer steps for solution. In this process the calculation that is needed can be performed without needing the variables while we are solving the problem. Moreover, it is a less error prone method compared to other methods available.

9. What are the requirements of synthetic division?

a) Divisor degree: 0 & Leading coefficient of divisor: 1

b) Divisor degree: 1 & Leading coefficient of divisor: 0

c) Divisor degree: 1 & Leading coefficient of divisor: 1

d) Divisor degree: 0 & Leading coefficient of divisor: 0

View Answer

Explanation: The divisor of any given polynomial for synthetic division should be of the degree 1, that is it should be a linear factor. Moreover, the coefficient of the divisor variable should be also equal to 1.

10. What will be the remainder & quotient by synthetic division on the following: (5x^{3} – x^{2} + 6) / (x-4)?

a) 310 & 5x^{2} + 19x + 76

b) 210 & 7x^{2} + 17x – 26

c) 210 & 7x^{2} – 17x – 26

d) 210 & 7x^{2} + 17x + 26

View Answer

Explanation: Given,

To determine root divisor, we have to solve divisor equation x-4=0

Therefore, our root becomes x=4

Write coefficients of the dividend 5x^{3} – x^{2} + 6 to the right and our root 4 to the left

4 | 5 | -1 | 0 | 6 |

Step-1: Write down the first coefficient 5

4 | 5 | -1 | 0 | 6 |

5 |

Step-2: Multiply our root 4 by our last result 5 to get 20 [4 × 5=20]

4 | 5 | -1 | 0 | 6 |

20 | ||||

5 |

Step-3: Add new result 20 to the next coefficient of the dividend -1, and write down the sum 19, [(-1) +20=19]

4 | 5 | -1 | 0 | 6 |

20 | ||||

5 | 19 |

Step-4: Multiply our root 4 by our last result 19 to get 76 [4 × 19=76]

4 | 5 | -1 | 0 | 6 |

20 | 76 | |||

5 | 19 |

Step-5: Add new result 76 to the next coefficient of the dividend 0, and write down the sum 76,[0+76=76]

4 | 5 | -1 | 0 | 6 |

20 | 76 | |||

5 | 19 | 76 |

Step-6: Multiply our root 4 by our last result 76 to get 304 [4 × 76=304]

4 | 5 | -1 | 0 | 6 |

20 | 76 | 304 | ||

5 | 19 | 76 |

Step-7: Add new result 304 to the next coefficient of the dividend 6, and write down the sum 310, [6+304=310]

4 | 5 | -1 | 0 | 6 |

20 | 76 | 304 | ||

5 | 19 | 76 | 310 |

We have completed the table and have obtained the following coefficients

5, 19, 76, 310

All coefficients, except last one, are coefficients of quotient, last coefficient is remainder.

Thus, quotient is 5x^{2} + 19x + 76 and remainder is 310.

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