Hint: Recall degree of polynomial in order to find the number of zeroes of polynomial.
Question.1. Which of the following statements is correct?
(a) A polynomial of degree 2 has three zeroes
(b) A polynomial of degree 3 has two zeroes
(c) A polynomial of degree 4 has four zeroes
(d) A polynomial of degree 5 has ten zeroes
Question.2. Ravi claims that the polynomial p(x)=mx^{a}+x^{2b} has 4b zeroes. For Ravi’s claim to be correct, which of these must be true?
(a) a = 2b or a = 4b
(b) a = 2 or a = 4b
(c) m = 2b
(d) m = 4b
Ans.1. (c) A polynomial of degree 4 has four zeroes
Ans.2. (a) a = 2b or a = 4b
Hint: Analyse the graph of the polynomials in order to find the number of zeroes of polynomial.
Question.3. Which of the following graphs could be for the simple polynomial x^{2} ?
(a)
(b)
(c)
(d)
Question.4. Consider the expression x^{m^{2}-1}+3x^{\frac{m}{2}}, where m is a constant. For what value of m, will the expression be a cubic polynomial?
(a) -2
(b) -1
(c) 1
(d) 2
Ans.3. (b)
Ans.4. (d) 2
Hint: Compute zeroes of the polynomials in order to verify the relationship between zeroes and the coefficients.
Question.5. Consider the polynomial in z, p(z)=z^{4}-2z^{3}+3. What is the value of the polynomial at z=-1?
(a) 2
(b) 3
(c) 5
(d) 6
Question.6. The polynomial in x, is x^{2}+kx+5, where k is a constant. At x=2, the value of the polynomial is 15.
What is the value of the polynomial at x=5?
(a) 3
(b) 18
(c) 35
(d) 45
Ans.5. (d) 6
Ans.6. (d) 45
Hint: Compute the sum and product of zeroes of the polynomial in order to find the quadratic polynomial.
Question.7. Which of these is a zero of the polynomials P(y)=3y^{3}-16y-8?
(a) -8
(b) -2
(c) 0
(d) 2
Question.8. Given that m+2, where m is a positive integer, is a zero of the polynomial q(x)=x^{2}-mx-6.
Which of these is the value of m?
(a) 1
(b) 2
(c) 3
(d) 4
Ans.7. (b) -2
Ans.8. (a) 1
Hint: Divide the two given polynomials in order to verify the division algorithm.
Question.9. The polynomial q(z)=z^{3}-4z+a when divided by the polynomial (z-3) leaves remainder 5. What is the value of a?
(a) –10
(b) –3
(c) 3
(d) 10
Question.10. The polynomial p(x)=x^{m}+x, where m>1, when divided by (x-a), leaves remainder 6. Given that a is a positive integer, what is the value of m?
(a) 2
(b) 3
(c) 5
(d) 6
Ans.9. (a) –10
Ans.10. (a) 2
Hint: Divide the given polynomial with its known zero in order to find all the other zeroes of that polynomial.
Question.11. Which of these is a factor of the polynomial p(x)=x^{3}+4x+5 ?
(a) (x-1)
(b) (x+1)
(c) (x-2)
(d) (x+2)
Question.12. The polynomial (x-a), where a>0, is a factor of the polynomial q(x)=4\sqrt{2} x^{2}-\sqrt{2}. Which of these is a polynomial whose factor is (x-\frac{1}{a})?
(a) x^{2}+x+6
(b) x^{2}-5x+4
(c) x^{2}+4x-3
(d) x^{2}+x-6
Ans.11. (b) (x+1)
Ans.12. (d) x^{2}+x-6