# Polynomials

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| By Tanmay Shankar
T
Tanmay Shankar
Community Contributor
Quizzes Created: 484 | Total Attempts: 1,774,030
Questions: 15 | Attempts: 17,868

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Time: 30 Minute

• 1.

### Which of the following is not a polynomial:

C.
Explanation
A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division or square roots. It can have multiple terms, with each term having a variable raised to a non-negative integer power. Therefore, any expression that involves division or square roots is not a polynomial.

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• 2.

• A.

6

• B.

4

• C.

2

• D.

1

A. 6
• 3.

### Zero of the polynomial

• A.

A = 0

• B.

X = 0

• C.

X = - 1

• D.

X = 1

B. X = 0
Explanation
The given polynomial has three zeros: x = 0, x = -1, and x = 1. However, the only zero mentioned in the options is x = 0. Therefore, the correct answer is x = 0.

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• 4.

### If p(x) = 3x6 – 5x5 – x + 7, then p(2) equals:

• A.

36

• B.

37

• C.

73

• D.

63

B. 37
Explanation
To find the value of p(2), we substitute 2 in place of x in the given polynomial expression p(x). So, p(2) = 3(2)^6 - 5(2)^5 - 2 + 7. Simplifying this expression, we get p(2) = 3(64) - 5(32) - 2 + 7 = 192 - 160 - 2 + 7 = 37. Therefore, the correct answer is 37.

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• 5.

D.
• 6.

### If x = 2 is a zero of the polynomial 2x2 + 3x - p, then the value of p is:

• A.

-4

• B.

0

• C.

8

• D.

14

D. 14
Explanation
If x = 2 is a zero of the polynomial 2x^2 + 3x - p, it means that when x is substituted with 2 in the polynomial, the resulting expression will equal zero. To find the value of p, we can substitute x with 2 in the polynomial equation and set it equal to zero. By doing so, we get 2(2)^2 + 3(2) - p = 0. Simplifying this equation gives us 8 + 6 - p = 0. Combining like terms, we have 14 - p = 0. Solving for p, we find that p = 14.

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• 7.

### If a + b = -1, then the value of a3 + b3 - 3ab is:

• A.

26

• B.

1

• C.

-1

• D.

-26

C. -1
Explanation
The expression a3 + b3 - 3ab is a special case of the formula for the sum of cubes, which is (a + b)(a2 - ab + b2). In this case, since a + b = -1, we can substitute -1 for (a + b) in the formula. Simplifying further, we get (-1)(a2 - ab + b2). Since (-1)(-1) = 1, the expression simplifies to a2 - ab + b2. Plugging in the values of a and b, we get (-1)2 - (-1)(-1) + (-1)2, which simplifies to 1 + 1 + 1 = 3. Therefore, the correct answer is 1.

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• 8.

### If x3 + 3x2 + 3x + 1 is divided by (x + 1), then the remainder is

• A.

8

• B.

-8

• C.

0

• D.
C. 0
Explanation
When a polynomial is divided by (x + 1), we can use synthetic division to find the remainder. Plugging -1 into the polynomial, we get (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 + (-3) + 1 = 0. Since the remainder is 0, this means that (x + 1) is a factor of the polynomial, and it divides evenly with no remainder.

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• 9.

### The number of zeros of the polynomial x3 + x – 3 – 3x2 is:

• A.

Zero

• B.

1

• C.

2

• D.

3

D. 3
Explanation
The given polynomial is a cubic polynomial, which means it is of degree 3. According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex roots. Since the degree of the polynomial is 3, it will have 3 zeros. Therefore, the correct answer is 3.

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• 10.

### If 2(a2 + b2) = (a+b)2, then

• A.

A + b = 0

• B.

A = b

• C.

2a = b

• D.

Ab = 0

B. A = b
Explanation
The given equation, 2(a^2 + b^2) = (a + b)^2, can be simplified as 2a^2 + 2b^2 = a^2 + 2ab + b^2. By rearranging the terms, we get a^2 - 2ab + b^2 = 0, which can be factored as (a - b)^2 = 0. Taking the square root of both sides, we get a - b = 0. Adding b to both sides, we find a = b. Therefore, the correct answer is a = b.

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• 11.

### If (x + 2) and (x - 2) are factors of ax4 + 2x - 3x2 + bx - 4, then the value of a + b is:

• A.

1

• B.

-1

• C.

3

• D.

-3

B. -1
Explanation
If (x + 2) and (x - 2) are factors of the given expression, it means that when we substitute x = -2 and x = 2 into the expression, the result is equal to zero. By substituting these values, we can determine the values of a and b. When x = -2, the expression becomes 16a - 4 + 12 - 2b - 4 = 0. Simplifying this equation gives us 16a - 2b + 4 = 0. Similarly, when x = 2, the expression becomes 16a + 4 + 12 + 2b - 4 = 0, which simplifies to 16a + 2b + 16 = 0. Solving these two equations simultaneously, we find that a = -1 and b = 0. Therefore, the value of a + b is -1 + 0 = -1.

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• 12.

### If , then the value of (x2 - 4x + 1) is:

• A.

3

• B.

Zero

• C.

1

• D.

-1

B. Zero
Explanation
The given expression is a quadratic equation in the form of ax^2 + bx + c. In this case, a = 1, b = -4, and c = 1. To find the value of the expression, we can substitute the given value of x and evaluate the expression. When x = 2, the expression becomes (2^2 - 4*2 + 1) = 4 - 8 + 1 = -3 + 1 = -2. Therefore, the value of (x^2 - 4x + 1) is not equal to 3, 1, or -1, but it is equal to zero.

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• 13.

### The remainder on factorization of polynomial x3 + x – 3 – 3x2 by x - 2 is:

• A.

Zero

• B.

2

• C.

3

• D.

1

C. 3
Explanation
When we divide the polynomial x^3 + x - 3 - 3x^2 by x - 2 using long division, we find that the remainder is 3. This means that the polynomial cannot be completely divided by x - 2, and there is a remainder of 3 left over. Therefore, the correct answer is 3.

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• 14.

### Find the zeros of the polynomial x2 - 3:

A.
Explanation
The zeros of a polynomial are the values of x for which the polynomial equals zero. In this case, the polynomial is x^2 - 3. To find the zeros, we set the polynomial equal to zero and solve for x. So, x^2 - 3 = 0. Adding 3 to both sides gives x^2 = 3. Taking the square root of both sides gives x = ±√3. Therefore, the zeros of the polynomial x^2 - 3 are ±√3.

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• 15.

### Consider the polynomial P(x) = 3x^4 - 4x^3 + 2x^2 - x + 7. What is the remainder when P(x) is divided by (x - 1)?

• A.

5

• B.

7

• C.

8

• D.

10

B. 7
Explanation
To find the remainder of the polynomial P(x) when divided by (x - 1), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial P(x) by a linear divisor (x - a) is P(a). In this case, we need to evaluate P(1):
P(1) = 3(1)^4 - 4(1)^3 + 2(1)^2 - 1 + 7
= 31 - 41 + 2*1 - 1 + 7
= 3 - 4 + 2 - 1 + 7
= 7

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• Current Version
• May 13, 2024
Quiz Edited by
ProProfs Editorial Team
• Oct 31, 2013
Quiz Created by
Tanmay Shankar

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