A cubic polynomial is defined by f(x) = 4x3 - 7x - 3. Find the remainder when f(x) is divided by (x-2). Show that (2x+1) is a factor of f(x) and hence factorize f(x) completely.

Welcome back — been a while! Let's pick up where you left off. 1. The cubic polynomial f(x) is defined by f(x) = 4x^3 - 7x - 3. a) Find the remainder when f(x) is divided by (x-2). Step 1: Use the Remainder Theorem, which states that the remainder when f(x) is divided by (x-a) is f(a). Here, a=2. f(2) = 4(2)^3 - 7(2) - 3 Step 2: Calculate the value. f(2) = 4(8) - 14 - 3 f(2) = 32 - 14 - 3 f(2) = 18 - 3 f(2) = 15 The remainder is 15. b) Show that (2x+1) is a factor of f(x) and hence factorize f(x) completely. Step 1: Use the Factor Theorem. If (2x+1) is a factor, then f(-(1)/(2)) must be 0. f(-(1)/(2)) = 4(-(1)/(2))^3 - 7(-(1)/(2)) - 3 f(-(1)/(2)) = 4(-(1)/(8)) + (7)/(2) - 3 f(-(1)/(2)) = -(1)/(2) + (7)/(2) - 3 f(-(1)/(2)) = (6)/(2) - 3 f(-(1)/(2)) = 3 - 3 f(-(1)/(2)) = 0 Since f(-(1)/(2)) = 0, (2x+1) is a factor of f(x). Step 2: Perform polynomial division to find the other factor. Dividing 4x^3 - 7x - 3 by (2x+1) gives 2x^2 - x - 3. So, f(x) = (2x+1)(2x^2 - x - 3). Step 3: Factorize the quadratic expression 2x^2 - x - 3. 2x^2 - x - 3 = 2x^2 - 3x + 2x - 3 = x(2x - 3) + 1(2x - 3) = (x+1)(2x-3) Therefore, the complete factorization of f(x) is (2x+1)(x+1)(2x-3). c) Solve the equation 4^3 - 7 - 3 = 0, for 0 360^. Step 1: Let x = . The equation becomes 4x^3 - 7x - 3 = 0, which is f(x)=0. From part b), we know f(x) = (2x+1)(x+1)(2x-3). So, (2x+1)(x+1)(2x-3) = 0. This gives the solutions for x: • 2x+1 = 0 x = -(1)/(2) • x+1 = 0 x = -1 • 2x-3 = 0 x = (3)/(2) Step 2: Substitute back x = and solve for . • = -(1)/(2) The reference angle is = ^-1((1)/(2)) = 60^. Since is negative, is in the second or third quadrant. = 180^ - 60^ = 120^ = 180^ + 60^ = 240^ • = -1 = 180^ • = (3)/(2) There are no solutions for this case, as the range of is [-1, 1]. The solutions for are 120^, 180^, 240^. *d) If and are the roots of the equation x^2 - 4x + 2 = 0, find the equation whose roots are + (1)/() and $ +