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**1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.**

**(i) 4 x^{2} - 3x + 7**

**(ii) y^{2} + âˆš2**

**(iii) 3âˆš t + tâˆš2**

**(iv) y + 2/y**

**(v) x^{10} + y^{3} + t^{50}**

(i) 4*x*^{2} - 3*x* + 7

There is only one variable *x* with whole number power so this polynomial in one variable.

(ii) *y*^{2} + âˆš2

There is only one variable *y* with whole number power so this polynomial in one variable.

(iii) 3âˆš2 + *t*âˆš2

There is only one variable *t* but in 3âˆš*t* power of *t* is 1/2 which is not a whole number so 3âˆš*t* + *t*âˆš2 is not a polynomial.

(iv) *y* + 2/*y*

There is only one variable *y * but 2/*y* = 2*y*-1 so the power is not a whole number so *y* + 2/*y* is not a polynomial.

(v) *x*^{10} + *y*^{3} + *t*^{50}

There are three variable *x*, *y* and* t* and there powers are whole number so this polynomial in three variable.

**2. Write the coefficients of x^{2} in each of the following:**

(i) 2 + x^{2 }+ x

(ii) 2 - x^{2} + x^{3}

^{(iii) }

(iv) âˆš2*x* - 1**Answer**

(i) coefficients of *x*^{2} = 1

(ii) coefficients of *x*^{2} = -1

(iii) coefficients of *x*^{2} = Ï€/2

(iv) coefficients of *x*^{2} = 0

**3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.**

**Answer**

3*x*^{35} + 7 and 4*x*^{100}

**4. Write the degree of each of the following polynomials:**

**(i) 5 x^{3} + 4x^{2} + 7x **

**(ii) 4 â€“ y^{2} **

**(iii) 5 t â€“ âˆš7**

**(iv) 3**

**Answer**

(i) 5*x*^{3} has highest power in the given polynomial which power is 3. Therefore, degree of polynomial is 3.

(ii) â€“ *y*^{2} has highest power in the given polynomial which power is 2. Therefore, degree of polynomial is 2.

(iii) 5*t* has highest power in the given polynomial which power is 1. Therefore, degree of polynomial is 1.

(iv) There is no variable in the given polynomial. Therefore, degree of polynomial is 0.**5. Classify the following as linear, quadratic and cubic polynomial:**

**(i) x^{2 }+ x**

â–º Quadratic Polynomial

**(ii) x - x^{3}**

â–º Cubic Polynomial

**(iii) y + y^{2} + 4**

â–º Quadratic Polynomial

**(iv) 1 + x**

â–º Linear Polynomial

**(v) 3 t**

â–ºLinear Polynomial

**(vi) r^{2}**

â–º Quadratic Polynomial**(vii) 7 x^{3}**

â–º Cubic Polynomial

**Exercise 2.2 1. Find the value of the polynomial at 5**

**(ii) x = - 1 **

**(iii) x = 2**

**Answer**

**2. Find p(0), p(1) and p(2) for each of the following polynomials:**

**(i) p(y) = y^{2} - y +1**

**(ii) p(t) = 2 +t + 2t^{2} - t^{3}**

(iii) p(x) = x^{3}

**(iv) p(x) = (x - 1) (x + 1)**

**Answer**

**3. Verify whether the following are zeroes of the polynomial, indicated against them.**

**(i) p(x) = 3x 1, x = -1/3**

**(ii) p(x) = 5x - Ï€, x = 4/5**

**(iii) p(x) = x^{2} - 1, x = 1, -1**

**(iv) p( x) = (x + 1) (x - 2), x = -1, 2**

(v) p(x) = x^{2} , x = 0

**(viii) p(x) = 2x + 1, x = 1/2**

(i) If

At,

Therefore,

(ii) If *x* = 4/5 is a zero of polynomial *p*(*x*) = 5*x* - Ï€ then *p*(4/5) should be 0.

At, *p*(4/5) = 5(4/5) - Ï€ = 4 - Ï€

Therefore, *x* = 4/5 is not a zero of given polynomial *p*(*x*) = 5*x* - Ï€.

(iii) If *x* = 1 and *x* = -1 are zeroes of polynomial *p*(*x*) = *x*^{2} - 1, then *p*(1) and *p*(-1) should be 0.

At, *p*(1) = (1)^{2} - 1 = 0 and

At, *p*(-1) = (-1)^{2} - 1 = 0

Hence, *x* = 1 and -1 are zeroes of the polynomial *p*(*x*) = *x*^{2} - 1.

(iv) If *x* = -1 and *x* = 2 are zeroes of polynomial *p*(*x*) = (*x* + 1) (*x* - 2), then *p*( - 1) and (2) should be 0.

At, *p*(-1) = (-1 + 1) (-1 - 2) = 0 (-3) = 0, and

At, *p*(2) = (2 + 1) (2 - 2) = 3 (0) = 0

Therefore, *x* = -1 and *x* = 2 are zeroes of the polynomial *p*(*x*) = (*x* + 1) (*x* - 2).

(v) If *x* = 0 is a zero of polynomial *p*(*x*) = *x*^{2}, then *p*(0) should be zero.

Here, *p*(0) = (0)^{2} = 0

Hence, *x* = 0 is a zero of the polynomial *p*(*x*) = *x*^{2}.

(viii) If *x* = 1/2 is a zero of polynomial *p*(*x*) = 2*x* + 1 then *p*(1/2) should be 0.

At, *p*(1/2) = 2(1/2) 1 = 1 + 1 = 2

Therefore, *x* = 1/2 is not a zero of given polynomial *p*(*x*) = 2*x* + 1.**4. Find the zero of the polynomial in each of the following cases: (i) p(x) = x + 5 **

**(ii) p(x) = x - 5 **

**(iii) p(x) = 2x + 5**

(iv) p(x) = 3x - 2

**(v) p(x) = 3x **

**(vi) p(x) = ax, a â‰ 0**

(vii) p(x) = cx d, c â‰ 0, c, are real numbers.

**Answer**

**Exercises 2.3 1. Find the remainder when **

(i)

By long division,

Therefore, the remainder is 0.

(ii) *x* - 1/2

By long division,

Therefore, the remainder is 27/8.

(iii) *x*

Therefore, the remainder is 1.

(iv) *x* + Ï€

Therefore, the remainder is [1 - 3Ï€ 3Ï€^{2} - Ï€^{3}].

(v) 5 + 2*x*

Therefore, the remainder is -27/8.**2. Find the remainder when x^{3} - ax^{2} + 6x - a is divided by x - a.**

By Long Division,

Therefore, remainder obtained is 5*a *when *x*^{3} - *ax*^{2} + 6*x* - *a* is divided by *x* -* a*.

**3. Check whether 7 + 3 x is a factor of 3x^{3} + 7x.**

We have to divide 3

By Long Division,

As remainder is not zero so 7 + 3*x* is not a factor of 3*x*^{3} + 7*x*.