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) \(y + \dfrac{2}{y}\)
(v) \(x^{10} + y^3 + t^{50}\)
(i) and (ii) are polynomials in one variable.
(v) is a polynomial in three variables.
(iii) and (iv) are not polynomials because their variable exponents are not whole numbers.
Write the coefficients of \(x^2\) in each of the following:
(i) \(2 + x^2 + x\)
(ii) \(2 - x^2 + x^3\)
(iii) \(\dfrac{\pi}{2} x^2 + x\)
(iv) \(\sqrt{2}x - 1\)
(i) 1
(ii) -1
(iii) \(\dfrac{\pi}{2}\)
(iv) 0
Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Example binomial of degree 35: \(3x^{35} - 4\)
Example monomial of degree 100: \(\sqrt{2} \, y^{100}\)
Write the degree of each of the following polynomials:
(i) \(5x^3 + 4x^2 + 7x\)
(ii) \(4 - y^2\)
(iii) \(5t - \sqrt{7}\)
(iv) \(3\)
(i) 3
(ii) 2
(iii) 1
(iv) 0
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) \(7x^3\)
(i) quadratic
(ii) cubic
(iii) quadratic
(iv) linear
(v) linear
(vi) quadratic
(vii) cubic