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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