1.1 Change in Tandem: แบบฝึกหัด

Find the zeros of the polynomial function \( f(x) = (x+2)(x-3)(2x+1) \).

Find the values of \( A \) and \( B \) such that \( \frac{5x - 7}{x^2 - 3x + 2} = \frac{A}{x - 1} + \frac{B}{x - 2} \). What is the value of \( A + B \)?

Find the slant (oblique) asymptote of the rational function \( f(x) = \frac{x^2 + 3x + 1}{x - 2} \).

Find the vertical asymptote of the rational function \( f(x) = \frac{2x + 3}{x - 4} \).

Analyze the end behavior of the rational function \( g(x) = \frac{-3x^5 + 2x^2 - 1}{4x^3 + 5x + 7} \). Which statement correctly describes the limits as \( x \) approaches infinity and negative infinity?

Given the polynomial \( P(x) = x^3 + kx^2 - 4x + 12 \).
(a) If the remainder when \( P(x) \) is divided by \( x - 1 \) is 6, find the value of the constant \( k \).
(b) Using the value of \( k \) found in part (a), find the remainder when \( P(x) \) is divided by \( x + 2 \).