Graph each polynomial function. Factor first if the expression is not in factored form.f(x)=x^{3} + 3x^{2} - 13x - 15 PSZ

Tobias Ali 2020-11-08 Answered

Graph each polynomial function. Factor first if the expression is not in factored form. f(x)=x3 + 3x2  13x  15

 
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Macsen Nixon
Answered 2020-11-09 Author has 117 answers

The rational Root Theorem states that a rational root of a polynomial equation an xn + an  1 xn  1 + s˙c + a2 x2 + a1 x + a0 x0=0 with integer coeffcients is of the form pq weher p is a factor of the constant term, a0, and q is a factor of the leading coefficient, an
f(x)=x3 + 3x2  13x  15
p could equal any factors of 15
So, ± 1, ± 3, ± 5, ± 15 q could equal any foctors of 1 So, ± 1 Therefore, possible rational zeros are ± 1, ± 3, ± 5, ± 15 Since the polynomial is x3 + 3x2  13x  15 so, the cooefficient are 1, 3, -13, -15 Let 1 is an actual zero of the polynomial and use synthetic division image The remainder is 0 so, -1 is an actual zero and (x + 1) is a factor of x3 + 3x2  13x  15 The equatient is x2 + 2x  15
f(x)=(x + 1)(x2 + 2x  15)
x2 + 2x  15=(x  3)(x + 5) Factor the trinomial So, f(x)=(x + 1)(x  3)(x + 5) The function in factored form The function has three zeros -1, 3, and -5 So, the graph of f(x) crosses the x-axis at (-1, 0), (3, 0), and (-5, 0) To find the y-intercept, substitute 0 for x in f(x) f(x)=x3 + 3x2  13x  15
f(0)=03 + 3(0)2  13(0)  15 Substitute 0 for x = 15 So, the function f(x) crosses the y-axis at (0, -15) The leading coefficient is 1 Since the leading coefficient is positive and the function f(x) of degree 3 (odd degree) So, the end behavior is x  , f(x)  
x  , f(x)   See the graph below image

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