**Q1: What is a polynomial?**

Answer: A polynomial is an

**algebraic expression**, in which**no variables appear in denominators**or**under radical signs**, and all**variables**that do**appear**are**powers of positive integers**.**Exercise 2.1**

**Q1: Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.**

(i) 4x

^{2}– 3x + 7
(ii) y

^{2}+ √2
(iii) 3√t + t√2

(iv) y +2/y

(v) x

^{10}+ y^{3}+ t^{50}
Answer:

(i) 4x

^{2}– 3x + 7
Because polynomial has one variable 'x', yes, this expression is a polynomial in one variable x.

(ii) y

^{2}+ √2
yes, this expression is a polynomial in one variable y.

(iii) 3√t + t√2

The exponent of variable t is ½, which is not a whole number. The expression is not a polynomial in one variable.

(iv) y +2/y

The variable has power of -ve integer, the expression is not a polynomial in one variable.

(v) x

^{10}+ y^{3}+ t^{50}
This expression is a polynomial in 3 variables x, y, and t. So, it is not a polynomial in one variable.

**Q2. Write the coefficients of x2 in each of the following:**

(i) 2 + x

^{2}+ x
(ii) 2 – x

^{2}+ x^{3}
(iii) (π/2)x

^{2}+ x
(iv) √2.x − 1

Answer: The numerical values (including +ve or -ve signs) of the terms in a polynomial are called the coefficients of the polynomial.

(i) 2 + x

^{2}+ x
The coefficient of x

^{2}is 1.
(ii) 2 – x

^{2}+ x^{3}
The coefficient of x

^{2}is -1.
(iii) (π/2)x

^{2}+ x
The coefficient of x

^{2}is π/2
(iv) √2.x − 1 = 0.x

^{2}+ √2.x − 1
The coefficient of x

^{2}is 0.**Q3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.**

Answer: Binomial of degree 35 = 3x

^{35}+ x
Monomial of degree 100 = 37t

^{100}**Note**: Degree of a polynomial is the highest power of the variable in the polynomial. Monomial has only one term in it. Binomial has two terms in it.

**Q4. Write the degree of each of the following polynomials:**

(i) 5x

^{3}+ 4x^{2}+7x
(ii) 4 – y

^{2}(iii) 5t – √7 (iv) 3
Answer: Degree of a polynomial is the highest power of variable in the polynomial.

(i) Degree of polynomial f(x) = 5x

^{3}+ 4x^{2}+7x is 3.
(ii) Degree of polynomial f(y) = 4 – y

^{2}is 2
(iii) Degree of polynomial f(t) = 5t – √7 is 1

**Q5. Classify the following as linear, quadratic and cubic polynomials:**

(i) x

^{2}+ x
(ii) x – x

^{3}
(iii) y + y

^{2}+ 4
(iv) 1 + x

(v) 3t

(vi) r

^{2}
(vii) 7x

^{3}
Answer:

Degree of linear polynomial = 1

Degree of quadratic polynomial = 2

Degree of cubic polynomial = 3

(i) x

^{2}+ x is a quadratic polynomial (degree = 2)
(ii) x – x

^{3}is a cubic polynomial. (degree = 3).
(iii) y + y

^{2}+ 4 is a quadratic polynomial (degree = 2)
(iv) 1 + x is linear polynomial (degree =1 )

(v) 3t is a linear polynomial (degree =1 )

(vi) r

^{2}is a quadratic polynomial (degree = 2)
(vii) 7x

^{3}is a cubic polynomial. (degree = 3).**Q6: What is the degree of a non-zero constant polynomial?**

Answer: Zero

**Q7: Polynomial having three terms are called as ____.**

Answer: Trinomials.

**Q8: What is the degree of the zero polynomial?**

Answer: The degree of the zero polynomial is not defined.

**Q9: The constant polynomial 0 is called _____.**

Answer: The zero polynomial.

**Q10: What is the standard form of writing a polynomial?**

Answer:

- While writing a polynomial in standard form, term with higher terms are written first.
- The coefficient of the first term (called leading coefficient) should be +ve.

**Q11: Write the polynomial 7x**

^{2}+ 8x + 6 + 5x^{3}in standard form.
Answer: Standard form 5x

^{3}+ 7x^{2}+ 8x + 6.**Q12: Write the polynomial 3x**

^{2}- 8x + 6 - 5x^{3}in standard form.
Answer: Write with terms in decreasing degree = -5x

Since the leading co-efficient be +ve, multiple all the terms with -1, i.e.

= -1 (-5x

= 5x

^{3}+ 3x^{2}- 8x + 6.Since the leading co-efficient be +ve, multiple all the terms with -1, i.e.

= -1 (-5x

^{3}+ 3x^{2}- 8x + 6)= 5x

^{3}- 3x^{2}+ 8x - 6
Never thought maths so easy

ReplyDelete