NCERT Solutions for Class 9 Math Chapter 2.1 POLYNOMIALS ALL SOLUTION

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EXERCISE 2.2 EXERCISE 2.3 EXERCISE 2.4 EXERCISE 2.5

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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)     `\4x^2-3x+7\ `

(ii)    `\y^2+\sqrt{2}\ `

(iii)    `\3\sqrt{t}+t\sqrt{2}\ `

(iv)    `\y+\sqrt{2}\ `

(v)    `\x^10+y^3+t^50\ `

solution :

(i)    `\4x^2-3x+7\ `

    The equation `\4x^2-3x+7\ ` can be written as `\4x^2-3x^1+7x^0\ `

    Since `\x\ ` is the only variable in the given equation and the powers of `\x ( 2, 1 and 0 )\ ` are whole numbers.

    we can say that the expression `\4x^2-3x+7\ ` is a polynomial in one variable.

(ii)    `\y^2+\sqrt{2}\ `

    The equation `\y^2+\sqrt{2}\ ` can be written as `\y^2+\sqrt{2}y^0\ `

    Since y is the only variable in the given equation and the powers of `\y ( 2 and 0 )\ ` are whole numbers.

     we can say that the expression `\y^2+\sqrt{2}\ ` is a polynomial in one variable.

(iii)    `\3\sqrt{t}+t\sqrt{2}\ `

    The equation `\3\sqrt{t}+t\sqrt{2}\ ` can be written as `\3t^\frac{1}{2}+t^1\sqrt{2}\ `

    Since, t is the only variable in the given equation and the powers of t `\( 1 and \frac{1}{2} )\ `.

    but, `\(\frac{1}{2})\ ` is not a whole number.

    Hence, we can say that the expression `\3\sqrt{t}+t\sqrt{2}\ ` is not a polynomial in one variable.

(iv)    `\y+\frac{2}{y}\ `

    The equation `\y+\frac{2}{y}\ ` an be written as `\y^1+2y^-1\ `

    Since, `\y\ ` is the only variable in the given equation and the powers of `\y ( 1 and -1)\ `.

    but, ( −1) is not a whole number.

    Hence, we can say that the expression `\y+\frac{2}{y}\ ` is not a polynomial in one variable.

(v)    `\x^10+y^3+t^50\ `

    Here, in the equation `\x^10+y^3+t^50\ `

    Since, `\( x, y, t )\ ` is the three variables in the given equation and the powers ( 10, 3, 50 ) are whole numbers.

    but, there are 3 variables used in the expression `\x^10+y^3+t^50\ `.

    Hence, it is not a polynomial in one 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)    `\frac{\pi}{2}x^2+x\ `

(iv)    `\sqrt{2}x-1\ `

solution :

(i)    `\2+x^2+x\ `

    The equation `\2+x^2+x\ ` can be written as `\2+1x^2+x^1\ `

    We know that,

    Coefficient is the number which multiplies the variable.

    Here, the number that multiplies the variable `\x^2\ ` is 1.

    the coefficients of `\x^2\ ` in `\2+x^2+x\ ` is 1.

(ii)    `\2−x^2+x^3\ `

    The equation `\2−x^2+x^3\ ` can be written as `\2+( −1 )x^2+x^3\ `.

    We know that,

    Coefficient is the number which multiplies the variable.

    Here, the number that multiplies the variable `\x^2\ ` is `\−1\ `.

    the coefficients of `\x^2\ ` in `\2−x^2+x^3\ ` is `\−1\ `.

(iii)    `\frac{\pi}{2}x^2+x\ `

    The equation `\frac{\pi}{2}x^2+x\ ` can be written as `\frac{\pi}{2}x^2+x^1\ `.

    We know that,

    Coefficient is the number which multiplies the variable.

    Here, the number that multiplies the variable `\x^2\ ` is ` \frac{\pi}{2}.

    the coefficients of `\x^2\ ` in `\frac{\pi}{2}x^2+x\ ` is `\frac{\pi}{2}\ `.

(iv)    `\sqrt{2}x-1\ `

    The equation `\sqrt{2}x-1\ ` can be written as `\0x^2+\sqrt{2}x-1\ ``\[ ∵ 0x^2=0]\ `

    We know that,

    Coefficient is the number which multiplies the variable.

    Here, the number that multiplies the variable `\x^2\ ` is 0.

    the coefficients of `\x^2\ ` in `\sqrt{2}x-1\ ` is 0.

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

solution :

    Binomial of degree 35:

    `\=3x^35+5\ `.

    Monomial of degree 100:

    `\=4x^100\ `.

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

(i)    `\5x^3+4x^2+7x\ `

(ii)    `\4-y^2\ `

(iii)    `\5t-\sqrt{7}\ `

(iv)    `\3\ `

solution :

(i)    `\5x^3+4x^2+7x\ `

    `\5x^3+4x^2+7x\ ` = `\5x^3+4x^2+7x^1\ `

    The powers of the variable `\x\ ` are: ( 3, 2, 1 ).

    the degree of `\5x^3+4x^2+7x\ ` is 3.

    becaus, 3 is the highest power of `\x\ ` in the equation.

(ii)    `\4-y^2\ `

    `\4-y^2\ `,

    The power of the variable `\y\ ` is 2

    the degree of `\4-y^2\ ` is 2.

    becaus, 2 is the highest power of `\y\ ` in the equation.

(iii)    `\5t-\sqrt{7}\ `

    `\5t-\sqrt{7}\ `,

    The power of the variable `\t\ ` is 1.

    the degree of `\5t^1-\sqrt{7}\ ` is 1.

    becaus, 1 is the highest power of `\t\ ` in the equation.

(iv)    `\3\ `

    `\= 3\ `

    `\= 3 × 1\ `

    `\= 3 × x^0\ `

    The power of the variable `\x\ ` is `\0\ `

    the degree of `\3\ ` is `\0\ `.

    becaus, `\0\ ` is the highest power of `\x\ ` in the equation.

5.    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\ `

solution :

(i)    `\x^2+x\ `

    The highest power of `\x^2+x\ ` is `\2\ `

    the degree of `\x^2+x\ ` is `\2\ `.

    Hence, `\x^2+x\ ` is a quadratic polynomial.

(ii)    `\x-x^3\ `

    The highest power of `\x-x^3\ ` is `\3\ `

    the degree of `\x-x^3\ ` is `\3\ `

    Hence, `\x-x^3\ ` is a cubic polynomial

(iii)    `\y+y^2+4\ `

    The highest power of `\y+y^2+4\ ` is `\2\ `

    the degree of `\y+y^2+4\ ` is `\2\ `

    Hence, `\y+y^2+4\ ` is a quadratic polynomial

(iv)    `\1+x\ `

    The highest power of `\1+x\ ` is `\1\ `

    the degree of `\1+x\ ` is `\1\ `

    Hence, `\1+x\ ` is a linear polynomial.

(v)    `\3t\ `

    The highest power of `\3t\ ` is `\1\ `.

    the degree of `\3t\ ` is `\1\ `.

    Hence, `\3t\ ` is a linear polynomial.

(vi)    `\r^2\ `

    The highest power of `\r^2\ ` is `\2\ `.

    the degree of `\r^2\ ` is `\2\ `

    Hence, `\r^2\ ` is a quadratic polynomial.

(vii)    `\7x^3\ `

    The highest power of `\7x^3\ ` is `\3\ `.

    the degree of `\7x^3\ ` is `\3\ `.

    Hence, `\7x^3\ ` is a cubic polynomial.

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EXERCISE 2.2 EXERCISE 2.3 EXERCISE 2.4 EXERCISE 2.5