**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) (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.** (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