Let f_1 = -3 + 2x + 2x^2, f_2 = 5 + x^2, f_3 = -20 + 8x + 7x^2 and f_4 = 3. give the definition of span{f_1, f_2}. use definition to show that h = span{f_1, f_2} is a subspace of p_2(x). find a basis of h. is f_3 in h? find a basis of p_2(x) containing {f_1, f_2}. is {f_1, f_2, f_3, f_4} a basis of p_2(x)? give a brief explanation for your answer. let b = {f_1, f_2, f_3}. find [f_4]b. let c = {f_4, f_5 = 3f_1 + 2f_2, f_6 = 2f_1 - 3f_2}. find p_c leftarrow b.

answer: b. f(x) = (x + 2)^3 (x^2 - 7x + 3)^4

step-by-step explanation:

polynomials always have as many roots as their order. this is a 4th order polynomial- hence 4 roots!

complex roots of polynomials always come in pairs. the quadratic equation has the "+/-" in that will alwaysgenerate m+ni amd m-ni, so 7-2i must be a root of this polynomial equation.

the order of the numerator and denominator are both 2. so the horizontal asymptote is y=a2/b2 =7

verticle asymptotes are the forbidden x-values. f(x) becomes undefined at x=-3 because this value will cause a zero denominator.

a graph of r(x) will have vertical asymptotes at ±1 because these values will make the denominator zero. so the domain of the function cannot include ±1.

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step-by-step explanation:

y - 25 = 2 ( x - 10 ) should be the answer