Polynomials

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| Written by Tanmay Shankar
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Tanmay Shankar
Community Contributor
Quizzes Created: 547 | Total Attempts: 1,801,734
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Polynomials - Quiz

Time: 30 Minute


Questions and Answers
  • 1. 

    Which of the following is not a polynomial:

    • A. 
    • B. 
    • C. 
    • D. 
    Correct Answer
    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. 

    The degree of polynomial .

    • A. 

      6

    • B. 

      4

    • C. 

      2

    • D. 

      1

    Correct Answer
    A. 6
  • 3. 

    Zero of the polynomial

    • A. 

      A = 0

    • B. 

      X = 0

    • C. 

      X = - 1

    • D. 

      X = 1

    Correct Answer
    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

    Correct Answer
    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. 

    The factorization of is:

    • A. 
    • B. 
    • C. 
    • D. 
    Correct Answer
    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

    Correct Answer
    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 x + y + 2 = 0, then x3 + y3 + 8 equals:

    • A. 

      (x + y + 2)3

    • B. 

      Zero

    • C. 

      6xy

    • D. 

      -6xy

    Correct Answer
    D. -6xy
    Explanation
    If x + y + 2 = 0, it means that the sum of x, y, and 2 is equal to zero. To find the value of x3 + y3 + 8, we can substitute the value of x + y + 2 into the expression. Therefore, (x + y + 2)3 = 0, and substituting this into the expression, we get 0 + 8 = 8. Hence, the correct answer is 8.

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  • 8. 

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

    • A. 

      26

    • B. 

      1

    • C. 

      -1

    • D. 

      -26

    Correct Answer
    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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  • 9. 

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

    • A. 

      8

    • B. 

      -8

    • C. 

      0

    • D. 
    Correct Answer
    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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  • 10. 

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

    • A. 

      Zero

    • B. 

      1

    • C. 

      2

    • D. 

      3

    Correct Answer
    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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  • 11. 

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

    • A. 

      A + b = 0

    • B. 

      A = b

    • C. 

      2a = b

    • D. 

      Ab = 0

    Correct Answer
    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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  • 12. 

    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

    Correct Answer
    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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  • 13. 

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

    • A. 

      3

    • B. 

      Zero

    • C. 

      1

    • D. 

      -1

    Correct Answer
    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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  • 14. 

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

    • A. 

      Zero

    • B. 

      2

    • C. 

      3

    • D. 

      1

    Correct Answer
    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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  • 15. 

    Find the zeros of the polynomial x2 - 3:

    • A. 
    • B. 
    • C. 
    • D. 
    Correct Answer
    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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