Can you explain what the following attached statement on cubic factors means. Thus, if the repeated root is -5/2, the remaining root is -1/(-5/2)^2 i.e -4/25.''  can you show how they got the remaining root and the sum of the roots. Also how did they decide the roots must be 1/2, 1/2 and 4.   

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K nas a repeated Dinerentiating, f(x) = 3x2 + 6x - 9 = 3(x + 2x - 3) = 3(x- 1)(x+3) the repeated factor of f(x) is (x- 1) or (x+ 3). If (x - 1) is a factor of f(x), then f(1) = 0 i.e. 1+3 - 9 +k = 0 giving k 5. If (x + 3) is a factor of f(x), then f(-3) 0 i.e. -27 + 27 + 27 + k = 0 giving k -27. Thus the possible values of k are 5 and -27. hen dealing with polynomial equations it may be helpful to use the rela- onships between roots and coefficients. In the case of the cubic equation + bx? + cx +d = 0 with roots a, B and y, x + B +y = aß + By + ya = a aB = -- milar results can be obtained for equations of higher degree. x-*)(-)(2-) xample 3 Find the roots of the equation 4x + 12x? - 15x + 4 = 0 given that it as a repeated root. f(x) = 4x + 12x2 - 15x +4 then f'(x) = 12x² + 24x – 15 Let = 3(4x? + 8x - 5) = 3(2x – 1)(2x + 5) %3D %3D Any repeated root of the equation f(x) = 0 is also a root of the equation f(x) = 0 * the repeated root must be either -5/2 or 1/2. [It is possible to decide which of these values is the required root by evaluating f(- 5/2) and f(1/2). However, the method given below may be quicker.] The product of the roots of f(x) = 0 is 4/4 i.d-1. Thus, if the repeated root is -5/2, the remaining root is-1+(-5/2)* i.e. -4/25. If the repeated root is 1/2, the remaining root is -1+(1/2) i.e. -4. Since the sum of the roots of f(x) = 0 is -12/4 i.e.-3, the roots must be 1/2, 1/2, - 4. %3D K nas a repeated Dinerentiating, f(x) = 3x2 + 6x - 9 = 3(x + 2x - 3) = 3(x- 1)(x+3) the repeated factor of f(x) is (x- 1) or (x+ 3). If (x - 1) is a factor of f(x), then f(1) = 0 i.e. 1+3 - 9 +k = 0 giving k 5. If (x + 3) is a factor of f(x), then f(-3) 0 i.e. -27 + 27 + 27 + k = 0 giving k -27. Thus the possible values of k are 5 and -27. hen dealing with polynomial equations it may be helpful to use the rela- onships between roots and coefficients. In the case of the cubic equation + bx? + cx +d = 0 with roots a, B and y, x + B +y = aß + By + ya = a aB = -- milar results can be obtained for equations of higher degree. x-*)(-)(2-) xample 3 Find the roots of the equation 4x + 12x? - 15x + 4 = 0 given that it as a repeated root. f(x) = 4x + 12x2 - 15x +4 then f'(x) = 12x² + 24x – 15 Let = 3(4x? + 8x - 5) = 3(2x – 1)(2x + 5) %3D %3D Any repeated root of the equation f(x) = 0 is also a root of the equation f(x) = 0 * the repeated root must be either -5/2 or 1/2. [It is possible to decide which of these values is the required root by evaluating f(- 5/2) and f(1/2). However, the method given below may be quicker.] The product of the roots of f(x) = 0 is 4/4 i.d-1. Thus, if the repeated root is -5/2, the remaining root is-1+(-5/2)* i.e. -4/25. If the repeated root is 1/2, the remaining root is -1+(1/2) i.e. -4. Since the sum of the roots of f(x) = 0 is -12/4 i.e.-3, the roots must be 1/2, 1/2, - 4. %3D