Q1. Which of the following expressions are polynomials in one variable and which are not?

State the reasons for your answers

- \(3x^{2} – 4x + 15\)
- \(y^{2} + 2\sqrt{3}\)
- \(3\sqrt{x} + \sqrt{2}x\)
- \(x – \frac{4}{x}\)
- \(x^{12} + y^{2} + t^{50}\)

Sol :

- \(3x^{2} – 4x + 15\)
– it is a polynomial of x - \(y^{2} + 2\sqrt{3}\)
– it is a polynomial of y - \(3\sqrt{x} + \sqrt{2}x\)
– it is not a polynomial since the exponent of \(3\sqrt{x}\) is not a positive term - \(x-\frac{4}{x}\)
– it is not a polynomial since the exponent of – \(\frac{4}{x}\) is not a positive term - \(x^{12} + y^{2} + t^{50}\)
– it is a three variable polynomial which variables of x, y, t

Q2. Write the coefficients of \(x^{2}\)

- \(17 – 2x + 7x^{2}\)
- \(9 – 12x + x^{2}\)
- \(\frac{\prod }{6}x^{2} – 3x + 4\)
- \(\sqrt{3}x – 7\)

Sol :

Given , to find the coefficients of \(x^{2}\)

- \(17 – 2x + 7x^{2}\)
– the coefficient is 7 - \(9 – 12x + x^{2}\)
– the coefficient is 0 - \(\frac{\prod }{6}x^{2} – 3x + 4\)
– the coefficient is \(\frac{\prod }{6}\) - \(\sqrt{3}x – 7\)
– the coefficient is 0

Q3. Write the degrees of each of the following polynomials :

- \(7x^{3} + 4x^{2} – 3x + 12\)
- \(12 – x + 2x^{2}\)
- \(5y – \sqrt{2}\)
- \(7- 7x^{0}\)
- 0

Sol :

Given , to find degrees of the polynomials

Degree is highest power in the polynomial

- \(7x^{3} + 4x^{2} – 3x + 12\)
– the degree is 3 - \(12 – x + 2x^{3}\)
– the degree is 3 - \(5y – \sqrt{2}\)
– the degree is 1 - \(7- 7x^{0}\)
– the degree is 0 - 0 – the degree of 0 is not defined

Q4. Classify the following polynomials as linear, quadratic, cuboc and biquadratic polynomials :

- \(x + x^{2} + 4\)
- 3x – 2
- \(2x + x^{2}\)
- 3y
- \(t^{2} + 1\)

f . \(7t^{4} + 4t^{2} + 3t – 2\)

Sol :

Given

- \(x + x^{2} + 4\)
– it is a quadratic polynomial as its degree is 2 - 3x – 2 – it is a linear polynomial as its degree is 1
- \(2x + x^{2}\)
– it is a quadratic polynomial as its degree is 2 - 3y – it is a linear polynomial as its degree is 1
- \(t^{2} + 1\)
– it is a quadratic polynomial as its degree is 2

f . \(7t^{4} + 4t^{2} + 3t – 2\)

Q5. Classify the following polynomials as polynomials in one variables, two – variables etc :

- \(x^{2} – xy + 7y^{2}\)
- \(x^{2} – 2tx + 7t^{2} – x + t\)
- \(t^{3} – 3t^{2} + 4t – 5\)
- xy + yz + zx

Sol :

Given

- \(x^{2} – xy + 7y^{2}\)
– it is a polynomial in two variables x and y - \(x^{2} – 2tx + 7t^{2} – x + t\)
– it is a polynomial in two variables x and t - \(t^{3} – 3t^{2} + 4t – 5 \)
– it is a polynomial in one variable t - \(xy + yz + zx\)
– it is a polynomial in 3 variables in x , y and z

Q6. Identify the polynomials in the following :

- \(f(x) = 4x^{3} – x^{2} -3x + 7\)
- b . \(g(x) = 2x^{3} – 3x^{2} + \sqrt{x} – 1\)
- \(p(x) = \frac{2}{3}x^{2} + \frac{7}{4}x + 9\)
- \(q(x) = 2x^{2} – 3x + \frac{4}{x} + 2\)
- \(h(x) = x^{4} – x^{\frac{3}{2}} + x – 1\)
- \(f(x) = 2 + \frac{3}{x} + 4x\)

Sol :

Given

- \(f(x) = \(4x^{3} – x^{2} -3x + 7\)
\(4x^{3} – x^{2} -3x + 7\]”> – it is a polynomial - b . \(g(x) = 2x^{3} – 3x^{2} + \sqrt{x} – 1\)
– it is not a polynomial since the exponent of \(\sqrt{x}\) is a negative integer - \(p(x) = \(\frac{2}{3}x^{2} + \frac{7}{4}x + 9\)
\(\frac{2}{3}x^{2} + \frac{7}{4}x + 9\]”> – it is a polynomial as it has positive integers as exponents - \(q(x) = 2x^{2} – 3x + \frac{4}{x} + 2\)
– it is not a polynomial since the exponent of \(\frac{4}{x}\) is a negative integer - h(x) = \(x^{4} – x^{\frac{3}{2}} + x – 1\)
– it is not a polynomial since the exponent of – \(x^{\frac{3}{2}}\) is a negative integer - f(x) = 2 + \(\frac{3}{x} + 4x\)
– it is not a polynomial since the exponent of \(\frac{3}{x}\) is a negative integer

Q7. Identify constant , linear , quadratic abd cubic polynomial from the following polynomials :

- \(f(x) = 0\)
- \(g(x) = 2x^{3} – 7x + 4\)
- \(h(x) = -3x + \frac{1}{2}\)
- \(p(x) = 2x^{2} – x + 4\)
- \(q(x) = 4x + 3\)
- \(r(x) = 3x^{3} + 4x^{2} + 5x – 7\)

Sol :

Given ,

- \(f(x) = 0\)
– as 0 is constant , it is a constant variable - \(g(x) = 2x^{3} – 7x + 4\)
– since the degree is 3 , it is a cubic polynomial - \(h(x) = -3x + \frac{1}{2}\)
– since the degree is 1 , it is a linear polynomial - \(p(x) = 2x^{2} – x + 4\)
– since the degree is 2 , it is a quadratic polynomial - \(q(x) = 4x + 3\)
– since the degree is 1 , it is a linear polynomial - \(r(x) = 3x^{3} + 4x^{2} + 5x – 7\)
– since the degree is 3 , it is a cubic polynomial

Q8. Give one example each of a binomial of degree 25, and of a monomial of degree 100

Sol :

Given , to write the examples for binomial and monomial with the given degrees

Example of a binomial with degree 25 – \(7x^{35} – 5\)

Example of a monomial with degree 100 – \(2t^{100}\)