Determine which of the following polynomials has \((x + 1)\) as a factor:
(i) \(x^3 + x^2 + x + 1\)
(ii) \(x^4 + x^3 + x^2 + x + 1\)
(iii) \(x^4 + 3x^3 + 3x^2 + x + 1\)
(iv) \(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)
(x + 1) is a factor of (i), but not a factor of (ii), (iii), and (iv).
Use the Factor Theorem to determine whether \(g(x)\) is a factor of \(p(x)\) in each case:
(i) \(p(x) = 2x^3 + x^2 - 2x - 1\), \(g(x) = x + 1\)
(ii) \(p(x) = x^3 + 3x^2 + 3x + 1\), \(g(x) = x + 2\)
(iii) \(p(x) = x^3 - 4x^2 + x + 6\), \(g(x) = x - 3\)
(i) Yes
(ii) No
(iii) Yes
Find the value of \(k\), if \(x - 1\) is a factor of \(p(x)\) in each of the following cases:
(i) \(p(x) = x^2 + x + k\)
(ii) \(p(x) = 2x^2 + kx + \sqrt{2}\)
(iii) \(p(x) = kx^2 - \sqrt{2}x + 1\)
(iv) \(p(x) = kx^2 - 3x + k\)
(i) -2
(ii) \(-(2 + \sqrt{2})\)
(iii) \(\sqrt{2} - 1\)
(iv) \(\dfrac{3}{2}\)
Factorise the following:
(i) \(12x^2 - 7x + 1\)
(ii) \(2x^2 + 7x + 3\)
(iii) \(6x^2 + 5x - 6\)
(iv) \(3x^2 - x - 4\)
(i) \((3x - 1)(4x - 1)\)
(ii) \((x + 3)(2x + 1)\)
(iii) \((2x + 3)(3x - 2)\)
(iv) \((x + 1)(3x - 4)\)
Factorise the following:
(i) \(x^3 - 2x^2 - x + 2\)
(ii) \(x^3 - 3x^2 - 9x - 5\)
(iii) \(x^3 + 13x^2 + 32x + 20\)
(iv) \(2y^3 + y^2 - 2y - 1\)
(i) \((x - 2)(x - 1)(x + 1)\)
(ii) \((x + 1)(x + 1)(x - 5)\)
(iii) \((x + 1)(x + 2)(x + 10)\)
(iv) \((y - 1)(y + 1)(2y + 1)\)