Hint: Recognize variables and their degree in a given algebraic expression in order to differentiate whether given expression is a polynomial in one variable or not.
Question.1. Consider the expression x^{m-1}+3; where m is a constant. What is the least integer value of m for which the given expression is a polynomial in one variable?
(a) 0
(b) 1
(c) 2
(d) 3
Question.2. Which of these is a polynomial in one variable?
(a) The perimeter of a square whose side length is represented by the expression \sqrt{x}.
(b) The area of a square whose side length is represented by the expression 1+\sqrt{x}.
(c) The area of a rectangle whose side lengths are represented by the expression 2+\sqrt{x} and \sqrt{x}.
(d) The perimeter of a rectangle whose side lengths are represented by the expression x^{2}+\sqrt{x} and 5-\sqrt{x}.
Ans.1. (c) 2
Ans.2. (d) The perimeter of a rectangle whose side lengths are represented by the expression x^{2}+\sqrt{x} and 5-\sqrt{x}.
Hint: Identify the degree of a given polynomial in order to classify an expression as zero, linear, quadratic and cubic polynomials.
Question.3. Consider the polynomials shown:
x^{2}+2x, 4x, 2x^{2}-3x+5, 0, \frac{3}{2}x-\frac{1}{2}
Which of the following tables correctly classifies the given polynomials as zero, linear, quadratic and cubic polynomials?
(a) 
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. (a)
Ans.4. (d) 2
Hint: Substitute the value of 'a' in a given expression p(x) in order to find the value of polynomial at 'a' i.e. p(a).
Question.5. Consider the polynomial in 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 value of the polynomial in x, is x^{2}+kz+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: Use given values for the variable 'x' in a polynomial p(x) in order to identify if the given value is a zero of the polynomials.
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 root 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: Using Remainder Theorem calculate division of p(x) by a linear polynomial 'x-a' in order to find that the remainder is p(a) and verify using long division method.
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: Apply factor theorem in order to determine if a linear polynomial 'x-a' is a factor of the given polynomial p(x).
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 \left(x-\frac{1}{a}\right)?
(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
Hint: Apply factor theorem in order to determine the value of an unknown constant 'k' in Polynomial P(x) when a linear polynomial x-a is a known factor of P(x).
Question.13. The polynomial (x-a) is a factor of the polynomial x^{4}-2x^{2}+kx+k, where k is a constant. Which of these is the correct relation between a and k?
(a) k=\frac{a^{2}(2-a^{2})}{1+a}
(b) k=\frac{a^{2}(2+a^{2})}{1+a}
(c) k=\frac{a^{2}(2+a^{2})}{1-a}
(d) k=\frac{a^{2}(2-a^{2})}{1-a}
Question.14. The polynomial (4x-3) is a factor of the polynomial q(x)=4x^{3}+x^{2}-11x+2r. What is the value of r?
(a) 2
(b) 3
(c) 4
(d) 11
Ans.13. (a) k=\frac{a^{2}(2-a^{2})}{1+a}
Ans.14. (b) 3
Hint: Apply factor theorem in order to factorize a given polynomial.
Question.15. The polynomial p(x)=x^{3}-5x^{2}-x+5 is such that p(1)=0 and p(-1)=0. Which of these is equivalent to p(x)?
(a) (x-1)(x+5)
(b) (x-1)(x+1)(x+5)
(c) (x-1)(x+1)(x-5)
(d) (x+1)(x-5)
Question.16. A polynomial p(x) of degree n is such that p(a)=0 and p(-b)=0. Which of the following is the factored form of the polynomial?
(a) (x-a)(x+b)g(x); where g(x) is a polynomial of degree n-2
(b) (x-a)(x+b)g(x); where g(x) is a polynomial of degree n
(c) (x+a)(x+b)g(x); where g(x) is a polynomial of degree n-2
(d) (x+a)(x+b)g(x); where g(x) is a polynomial of degree n
Ans.15. (c) (x-1)(x+1)(x-5)
Ans.16. (a) (x-a)(x+b)g(x); where g(x) is a polynomial of degree n-2
Hint: Factorize a given polynomial using splitting middle-term method and factor theorem in order to compare the results of the two.
Question.17. Which of these is obtained by factorizing the polynomial 10x^{2}-9x+2?
(a) (2x-1)(5x-2)
(b) (2x-1)(5x+2)
(c) (2x+1)(5x+2)
(d) (2x+1)(5x-2)
Question.18. The zeroes of the polynomial p(x)=x^{2}-(2k+1)x+16 are positive integers. Given that k is an integer, which of these is equivalent to the polynomial?
(a) (x-1)(x+16)
(b) (x-1)(x-16)
(c) (x-2)(x-8)
(d) (x-4)(x-4)
Ans.17. (a) (2x-1)(5x-2)
Ans.18. (b) (x-1)(x-16)
Hint: Point out to an algebraic identity that can be used in order to factorize a given expression.
Question.19. Which of these identities can be used to factorize the expression 4x^{2}-19x+16?
(a) (x-a)^{2}=x^2-2a+a^2
(b) (x+a)^{2}=x^2+2a+a^2
(c) (x-a)(x-b)=x^2-(a+b)x+ab
(d) (x-a)(x+a)=x^2-a^2
Question.20. The volume of a cube is given by the expression 27x^{3}+8y^{3}+54x^{2}y+36xy^{2}. What is the expression for the side length of the cube?
(a) 3x+2y
(b) 3x-2y
(c) 9x-8y
(d) 9x+8y
Ans.19. (c) (x-a)(x-b)=x^2-(a+b)x+ab
Ans.20. (a) 3x+2y
Hint: Select appropriate algebraic identities in order to evaluate the values of given expressions.
Question.21. Which of these identities can be used to find the value of the expression 97 \times 103?
(a) (x-y)^{2}=x^2-2y+y^2
(b) (x+y)^{2}=x^2+2y+y^2
(c) (x+y+z)^{2}=x^2+y^2+z^2+2xy+2yz+2xz
(d) (x-y)(x+y)=x^2-y^2
Question.22. Given that 100^{2}=a^{2}, which expression gives the value of the expression 103 \times 108?
(a) a^{2}+11a+24
(b) a^{2}+24a+11
(c) a^{2}+24a+24
(d) a^{2}+11a+11
Ans.21. (d) (x-y)(x+y)=x^2-y^2
Ans.22. (a) a^{2}+11a+24
