Let p : R→ R be a non-constant polynomial, that is, for every x in R, p(x) = α0 + α1x+ ... + anx, with an + 0. Prove that, if n is even then lim→∞ p(x) = +∞ and limx→-∞ p(x) = +∞ when an > 0, and lim→∞ p(x) = · 0, and limÃ→∞ p(x) = −∞ and lim¸→-∞ p(x) = when an < 0 and n is odd.

2 1. Consider a polynomial function f(x) = co + C₁ x + c₂x² +...+ cd-1xd-1 + caxd. A. Applying the addition rule, the constant multiple rule, and the power rule for derivatives, we can compute that ƒ'(x) = c₁ + 2c2x + … + (d − 1)cd-1xd-2 + (d)caxd-¹. Explain. B. Following the general outline from Part A, compute the derivatives of each of the following functions: g(x) = 5x³ 10x² + 7x - 11. p(x) = −7x² – 3x² + πx √17. O O 2. Consider the function f(x) = (2x³ – 5x + 1) (3x² − 4). A. Distribute out the product of the two polynomials multiplied in the expression of f and simplify your result to find an equivalent expression for f(x).

1. A polynomial function is given by q(x) = x(x-2)(x+5). On which interval(s) is q(x) > 0? a) [-, -5] U [0,2] b) (-0, -5] U [0,2] c) [-5,0] U [2, +∞) d) [-5,2]