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Q.1. Which of the following expressions are polynomials? In case of a polynomial, write its degree.
(i) x^{5 }− 2x^{3 }+ x + √3
(ii) y^{3 }+ √3y
(iii)
(iv) x^{100} − 1
(v)
(vi) x^{−2} + 2x^{−1 }+ 3
(vii) 1
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv) 2x^{3 }+ 3x^{2 }+ √x − 1
Ans.
(i) x^{5} − 2x^{3} + x +√3 is an expression having only nonnegative integral powers of x. So, it is a polynomial. Also, the highest power of x is 5, so, it is a polynomial of degree 5.
(ii) y^{3 }+ √3y is an expression having only nonnegative integral powers of y. So, it is a polynomial. Also, the highest power of y is 3, so, it is a polynomial of degree 3.
(iii) is an expression having only nonnegative integral powers of t. So, it is a polynomial. Also, the highest power of t is 2, so, it is a polynomial of degree 2.
(iv) x^{100 }− 1 is an expression having only nonnegative integral power of x. So, it is a polynomial. Also, the highest power of x is 100, so, it is a polynomial of degree 100.
(v) is an expression having only nonnegative integral powers of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.
(vi) x^{−2}+2x^{−1 }+ 3 is an expression having negative integral powers of x. So, it is not a polynomial.
(vii) Clearly, 1 is a constant polynomial of degree 0.
(viii) Clearly, is a constant polynomial of degree 0.
(ix)
This is an expression having negative integral power of x i.e. −2. So, it is not a polynomial.
(x) is an expression having only nonnegative integral power of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.
(xi)is an expression having negative integral power of x. So, it is not a polynomial.
(xii)
In this expression, the power of x is 1/2 which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.
(xiii) is an expression having only nonnegative integral powers of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.
(xiv)
In this expression, one of the powers of x is 3/2 which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.
(xv)
In this expression, one of the powers of x is 1/2 which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.
Q.2. Identify constant, linear, quadratic, cubic and quartic polynomials from the following.
(i) –7 + x
(ii) 6y
(iii) –z^{3}
(iv) 1 – y – y^{3}
(v) x – x^{3} + x^{4}
(vi) 1 + x + x^{2}
(vii) – 6x^{2}
(viii) – 13
(ix) – p
Ans.
(i) –7 + x is a polynomial with degree 1. So, it is a linear polynomial.
(ii) 6y is a polynomial with degree 1. So, it is a linear polynomial.
(iii) –z^{3} is a polynomial with degree 3. So, it is a cubic polynomial.
(iv) 1 – y – y^{3} is a polynomial with degree 3. So, it is a cubic polynomial.
(v) x – x^{3} + x^{4 }is a polynomial with degree 4. So, it is a quartic polynomial.
(vi) 1 + x + x^{2} is a polynomial with degree 2. So, it is a quadratic polynomial.
(vii) – 6x^{2} is a polynomial with degree 2. So, it is a quadratic polynomial.
(viii) –13 is a polynomial with degree 0. So, it is a constant polynomial.
(ix) – p is a polynomial with degree 1. So, it is a linear polynomial.
Q.3. Write
(i) the coefficient of x^{3} in x+3x^{2 }− 5x^{3 }+ x^{4}.
(ii) the coefficient of x in √3−2√2x + 6x^{2}.
(iii) the coefficient of x^{2} in 2x – 3 + x^{3}.
(iv) the coefficient of x in
(v) the constant term in
Ans.
(i) The coefficient of x3 in x+3x^{2}−5x^{3 }+ x^{4} is − 5.
(ii) The coefficient of x in √3 − 2√2x + 6x^{2} is −2√2.
(iii) 2x – 3 + x^{3} = – 3 + 2x + 0x^{2 }+ x^{3}
The coefficient of x^{2 }in 2x – 3 + x^{3 }is 0.
(iv) The coefficient of x in
(v) The constant term in
Q.4. Determine the degree of each of the following polynomials.
(i)
(ii) y^{2}(y – y^{3})
(iii) (3x – 2) (2x^{3 }+ 3x^{2})
(iv)
(v) – 8
(vi) x^{–2}(x^{4} + x^{2})
Ans.
(i)
Here, the highest power of x is 2. So, the degree of the polynomial is 2.
(ii) y^{2}(y – y^{3}) = y^{3} – y^{5}
Here, the highest power of y is 5. So, the degree of the polynomial is 5.
(iii) (3x – 2)(2x^{3} + 3x^{2}) = 6x^{4 }+ 9x^{3} – 4x^{3} – 6x^{2} = 6x^{4} + 5x^{3} – 6x^{2}
Here, the highest power of x is 4. So, the degree of the polynomial is 4.
(iv)
Here, the highest power of x is 1. So, the degree of the polynomial is 1.
(v) – 8
–8 is a constant polynomial. So, the degree of the polynomial is 0.
(vi) x^{–2}(x^{4} + x^{2}) = x^{2} + x^{0} = x^{2} + 1
Here, the highest power of x is 2. So, the degree of the polynomial is 2
Q.5.
(i) Give an example of a monomial of degree 5.
(ii) Give an example of a binomial of degree 8.
(iii) Give an example of a trinomial of degree 4.
(iv) Give an example of a monomial of degree 0.
Ans.
(i) A polynomial having one term is called a monomial. Since the degree of required monomial is 5, so the highest power of x in the monomial should be 5.
An example of a monomial of degree 5 is 2x^{5}.
(ii) A polynomial having two terms is called a binomial. Since the degree of required binomial is 8, so the highest power of x in the binomial should be 8.
An example of a binomial of degree 8 is 2x^{8} − 3x.
(iii) A polynomial having three terms is called a trinomial. Since the degree of required trinomial is 4, so the highest power of x in the trinomial should be 4.
An example of a trinomial of degree 4 is 2x^{4} − 3x + 5.
(iv) A polynomial having one term is called a monomial. Since the degree of required monomial is 0, so the highest power of x in the monomial should be 0.
An example of a monomial of degree 0 is 5.
Q.6. Rewrite each of the following polynomials in standard form.
(i) x−2x^{2 }+ 8 + 5x^{3}
(ii) 2/3 + 4y^{2 }− 3y + 2y^{3}
(iii) 6x^{3 }+ 2x − x^{5 }− 3x^{2}
(iv) 2 + t − 3t^{3 }+ t^{4 }− t^{2}
Ans.
A polynomial written either in ascending or descending powers of a variable is called the standard form of a polynomial.
(i) 8+x−2x^{2}+5x^{3} is a polynomial in standard form as the powers of x are in ascending order.
(ii) 2/3 − 3y + 4y^{2}+2y^{3} is a polynomial in standard form as the powers of y are in ascending order.
(iii) 2x−3x^{2 }+ 6x^{3 }− x^{5} is a polynomial in standard form as the powers of x are in ascending order.
(iv) 2 + t − t^{2 }− 3t^{3}+ t^{4} is a polynomial in standard form as the powers of t are in ascending order.
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