Find the value of the polynomial \(5x - 4x^2 + 3\) at:
(i) \(x = 0\)
(ii) \(x = -1\)
(iii) \(x = 2\)
(i) 3
(ii) -6
(iii) -3
Find \(p(0)\), \(p(1)\), and \(p(2)\) for each of the following polynomials:
(i) \(p(y) = y^2 - y + 1\)
(ii) \(p(t) = 2 + t + 2t^2 - t^3\)
(iii) \(p(x) = x^3\)
(iv) \(p(x) = (x - 1)(x + 1)\)
(i) 1, 1, 3
(ii) 2, 4, 4
(iii) 0, 1, 8
(iv) -1, 0, 3
Verify whether the following are zeroes of the polynomial, as indicated:
(i) \(p(x) = 3x + 1\), \(x = -\dfrac{1}{3}\)
(ii) \(p(x) = 5x - \pi\), \(x = \dfrac{4}{5}\)
(iii) \(p(x) = x^2 - 1\), \(x = 1, -1\)
(iv) \(p(x) = (x + 1)(x - 2)\), \(x = -1, 2\)
(v) \(p(x) = x^2\), \(x = 0\)
(vi) \(p(x) = lx + m\), \(x = -\dfrac{m}{l}\)
(vii) \(p(x) = 3x^2 - 1\), \(x = -\dfrac{1}{\sqrt{3}}, \dfrac{2}{\sqrt{3}}\)
(viii) \(p(x) = 2x + 1\), \(x = \dfrac{1}{2}\)
(i) Yes
(ii) No
(iii) Yes
(iv) Yes
(v) Yes
(vi) Yes
(vii) \(-\dfrac{1}{\sqrt{3}}\) is a zero, but \(\dfrac{2}{\sqrt{3}}\) is not a zero
(viii) No
Find the zero of the polynomial in each of the following cases:
(i) \(p(x) = x + 5\)
(ii) \(p(x) = x - 5\)
(iii) \(p(x) = 2x + 5\)
(iv) \(p(x) = 3x - 2\)
(v) \(p(x) = 3x\)
(vi) \(p(x) = ax\), \(a \neq 0\)
(vii) \(p(x) = cx + d\), where \(c, d\) are real numbers
(i) -5
(ii) 5
(iii) -\dfrac{5}{2}
(iv) \dfrac{2}{3}
(v) 0
(vi) 0
(vii) -\dfrac{d}{c}