4. a. Without using technology, sketch the graphs for each function provided and state the following for each: . end behaviour (write these something like (DZ->4, don't use infinity statement) . y-intercept . X-intercept(s) . multiplicity of each factor Be sure to justify your answers mathematically. i. 3 marks f(x) = (x+2)(x-1)"2 ii. 3 marks g(x) = -a (x + 2)"3(x-1) iii. 4 marks h(x) = (x - a)(x - b)(x+c)2, where a>0, b>0, and c>0, 5a. 4 marks Completely factor the following polynomial. Show the use of the remainder/factor theorem, long division and trinomial factoring. P(x) = 2x4 +7x3 +x2 - 16x -12 b. 4 marks Solve the following polynomial equation. Show the use of the remainder/factor theorem and synthetic division and leave your answer in exact form. x3-7x2-45x-45=0 c. 2 marks When a polynomial function P(x) is divided by 2x - 3 the quotient is x2 - x - 3 and the remainder is 7. Find the polynomial function. 6a. 2 marks Given y = ax"5 + bx"4 + cx"3 + dx"2 + ex + f where a>0 and f4). Y-Intercept: * To find the y-intercept, set c = 0: f(0) - (0 + 2) . (0 -1)2 - 2 . 1=2 . Y-intercept: (0, 2) X-Intercepts: * To find the x-intercepts, set f(x) = 0\\: \\(0 = (x + 2) \\cdot (x - 1)~2 . Solve for : *+2-0-x - -2 (Multiplicity: 1) x-1=0- x = 1 (Multiplicity: 2) So, there are two x-intercepts: (-2, 0) and (1, 0). Multiplicity of Each Factor: . The factor (x + 2) has multiplicity 1. * The factor (x - 1) has multiplicity 2.ii. g(x) = -a . (x+2)3. (x-1) End Behavior: As @ approaches positive infinity, g (@ ) decreases without bound (Q 2->4). * As a approaches negative infinity, g( ) increases without bound (Q 2->4). Y-Intercept: . To find the y-intercept, set x = 0: 9(0) - -a . (0+2)3 . (0 -1) - -8a . Y-intercept: (0, -8a) X-Intercepts: * To find the x-intercepts, set g(x) = 0\\: (0 = -a \\cdot (x + 2)^3 \\cdot (x - 1) . Solve for x\\: \\(x+2=0= = -2 (Multiplicity: 3) x- 1= 0- x = 1 (Multiplicity: 1) So, there are two x-intercepts: (-2, 0) and (1, 0). Multiplicity of Each Factor: * The factor (x + 2) has multiplicity 3. . The factor (x - 1) has multiplicity 1. ii. h(x) = (x - a) (x - b) . (x + c)2, where a > 0, b > 0, and c >0 End Behavior: * As a approaches positive infinity, h(@) increases without bound (Q 2->4). " As a approaches negative infinity, h(@) increases without bound (Q 2->4).Problem 4b: Completely Factoring the Polynomial P(x) - 2x + 72 + 2 -16x -12 First, check for rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term divided by the factors of the leading coefficient: Possible rational roots: 1, 12, 13, 14, 16, +12 Now, perform synthetic division or polynomial long division to determine the quotient. After dividing, you will have a polynomial of lower degree: Synthetic division by a potential root, such as x = 1: 2 71 1 - 16 - 12 This yields a quotient of 2x + 9x- + 11x - 27. Now, check for rational roots of this new polynomial. You can find that a - 3 is a root. Perform synthetic division or polynomial long division again: 3 2911 - 27 This time, the quotient is 2x- + 15x + 33. Factor the quadratic 2x- + 15x + 33, which factors to (2x + 3) (x + 11). Now you have factored the polynomial: P(x) - 2x + 72 + 2' - 16x -12 -2(x -1)(x - 3)(x + 3)(2x + 3)(x + 11)Problem 4c: Solving the Polynomial Equation The polynomial equation is: x3 - 7x- - 45x - 45 - 0 You can use the Rational Root Theorem to find potential rational roots. The possible rational roots are the factors of the constant term (-45) divided by the factors of the leading coefficient (1): Possible rational roots: 11, 13, 15, 19, +15, 145 Now, use synthetic division or polynomial long division to check for roots. You will find that c = 3 is a root: 3 1 -7 -45 -45 This results in a quotient of x- - 10x - 15. Factor the quadratic - - 10x - 15, which factors to (x - 5) (x + 3). So, you have factored the polynomial equation as: x3- 7x2 - 45x - 45 - (x - 3) (x - 5) (x + 3) - 0Problem 4d: Finding the Polynomial Function You are given that 1when a polynomial function P[:3} is divided by 2:3 3, the quotient i5 :32 :3 3 and the remainder is T. To find P(L3) you can 1write it as the product of the diyieor, the quotient, and add the remainder: 13(13} [2:3 3:}[132 :3 3} 7 How, you haye the polynomial function PI[:3}. Problem 6a: Sketching the Graph Given V- a . x'+b . etc . x +d . x te . x + f wherea > 0, f 0: This tells us that the leading term of the polynomial is a . ", indicating that the graph rises to the right (positive infinity) and falls to the left (negative infinity). 2. f