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+ 4) Consider the polynomial p(x) = -2x – 5x +x* + 10x + 4x2 – 5x – 3, which can be written in factored form as p(x) = -(2x + 3)(x - 1)(x + 1)2 a. Describe the end behavior of the graph, including a sketch if you wish. Explain how you know. (2 pts) pex -1 -3 -2 D 2 3 The function points down to the left and down to the right. It is neither even nor odd. It has three roots and one y intercept. From the graph it has one local minima and one global maxima. b. Find all real roots of the polynomial. Explain how you know based on the equation. (3 pts) There are three roots to the equation, that is solve the equation when p(x) = 0. -2x6 - 5x5 + x* + 10x8 + 4x2 – 5x – 3 = 0 Or this can be deduced from the factored form of the equation, where we can more easily infer the values that make a Zero appear in any of the factors There are three roots values which are: -1.5,-1,1 c. Describe the behavior of the graph close to each of the zeroes. Explain how you can tell the graph has this behavior based on the factored form of the polynomial. (3 pts) From the above answer, when p(x) = 0, there are 3 x in intercepts, these can be deduced from the factored equation form above, which yields –(3/2),-1,1 The points are (-1.5,0)(-1,0),(1,0) When x=0, the p(x) intercept is -3, the answer is obtained by substituting for the value of x in the above factored equation, the resulting point is (0, -3)

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solution Consider Pex) = -2x6_5x5tah +10x3+4x2_5x-3 gro factored form Peal -(2x+3)(x-1)(x+1) given polynomial is f LS negative then the as x → 1 (a) end behaviour of the dipicted by the degree of the polynomial and the sign of its leading cofficient Pex) is even degree & its leading cofficient then the graph will begin & end in the downward direction ile Pca) → - 00 degree 6. Ceren) : The begin & end in the same direction, Next, the leading cofficient is -2 (negative) ... The graph will be downwards. So, the graph of Pexe) is same direction downwards. an our Case graph will getting off in the

(b) we have Pea) = -(2(x)+3)(x-1)=(x+1)3 het Pex)=o gives =0 -(2x+3)(x-1) ? Co2+1) = → -(2x+3) = 0 , (2-1) = 0 op (x+13=0 2xp320 xt or Cx+1)zo / x= -32 ,2=1, 1 er a=-1,-1,-) X = -37221;1,-1,-1, 6-1 Amsung these roots one x The roots are According to findamental Meerem the degree 6 polynomial has 6 Leros (roots) x=²-½ has multi pilicity has multiplianty two has multiplienty three (C) Remember that, At the zero (root) of odd multipilicity, the graph will cross x-axis at the zero (root) of even multip graph will bounce x-axis & not go through it. { x= 1 and licity the &

Case 23 multiplicity of is cro x= In our -3%=-1.5 is zero one Coold) → The graph x-axis at x=-1.5 Crossing is zero of multiplicity of 2 (even) The graph will bonce off the x-axis 2= y is hero of multiplicity 4 3(odd) at graph Crosses axis since it has multiplicnly graph flattened mere at = t the 3 the seems 2=