1. Perform Addition $(f + g)(x)$ where: $f(x) = 4x^4 - 8x^3 + 6x^1 - 5x^1 + 9x - 8$ $g(x) = 2x^4 + 3x^1 - 2x^1 + 4x^2 - 6x + 5$ Simplify terms first: $f(x) = 4x^4 - 8x^3 + (6x^1 - 5x^1) + 9x - 8 = 4x^4 - 8x^3 + x + 9x - 8 = 4x^4 - 8x^3 + 10x - 8$ $g(x) = 2x^4 + (3x^1 - 2x^1) + 4x^2 - 6x + 5 = 2x^4 + x + 4x^2 - 6x + 5$ Add terms by degree: $(4x^4 + 2x^4) + (-8x^3 + 0) + (0 + 4x^2) + (10x + x - 6x) + (-8 + 5)$ $= 6x^4 - 8x^3 + 4x^2 + 5x - 3$ 2. Perform Subtraction $(f - g)(x)$ where: $f(x) = 12x^2 - 8x^4 + 7x^1 - 11x^1 + 15x - 18$ $g(x) = 8x^1 - 3x^4 + 4x^1 - 8x^1 + 7x - 10$ Simplify terms: $f(x) = -8x^4 + 12x^2 + (7x - 11x) + 15x - 18 = -8x^4 + 12x^2 -4x + 15x - 18 = -8x^4 + 12x^2 + 11x -18$ $g(x) = (-3x^4) + (8x + 4x - 8x) + 7x -10 = -3x^4 + 4x + 7x -10 = -3x^4 + 11x -10$ Subtract term wise: $(-8x^4 - (-3x^4)) + (12x^2 - 0) + (11x - 11x) + (-18 - (-10))$ $= (-8x^4 + 3x^4) + 12x^2 + 0 + (-18 +10) = -5x^4 + 12x^2 - 8$ 3. Perform Division $(f / g)(x)$ where: $f(x) = 42x^1 + 4x^1 - 72x^1 + 11x^1 - 51x + 36$ $g(x) = 7x - 4$ Simplify numerator: $42x + 4x - 72x + 11x - 51x + 36 = (42 + 4 -72 + 11 -51)x + 36 = (-66)x + 36 = -66x + 36$ Division: $$\frac{-66x + 36}{7x - 4}$$ 4. Perform Division $(f / g)(x)$ where: $f(x) = 32x^4 - 40x^1 - 64x^1 + 24x^1 - 20x + 72$ $g(x) = 4x - 8$ Simplify numerator: $32x^4 + (-40 -64 + 24)x - 20x + 72 = 32x^4 + (-80)x - 20x + 72 = 32x^4 - 100x + 72$ Division: $$\frac{32x^4 - 100x + 72}{4x - 8}$$ 5. Perform Division $(f / g)(x)$ where: $f(x) = 56x^1 - 19x^4 - 43x^1 - 22x^1 - 26x + 40$ $g(x) = 7x - 5$ Simplify numerator: $(-19x^4) + (56x - 43x -22x - 26x) + 40 = -19x^4 + (-35x) + 40$ Division: $$\frac{-19x^4 - 35x + 40}{7x - 5}$$ --- Test III: Find Equations of Lines 1. Points (3,4) and (6,7) Calculate slope: $$m = \frac{7 - 4}{6 - 3} = \frac{3}{3} = 1$$ Use point-slope form with point (3,4): $$y - 4 = 1(x - 3)$$ $$y = x + 1$$ 2. Points (1,2) and (3,5) $$m = \frac{5 - 2}{3 - 1} = \frac{3}{2}$$ $$y - 2 = \frac{3}{2}(x - 1)$$ $$y = \frac{3}{2}x - \frac{3}{2} + 2 = \frac{3}{2}x + \frac{1}{2}$$ 3. Points (3,2) and (8,5) $$m = \frac{5 - 2}{8 - 3} = \frac{3}{5}$$ $$y - 2 = \frac{3}{5}(x - 3)$$ $$y = \frac{3}{5}x - \frac{9}{5} + 2 = \frac{3}{5}x + \frac{1}{5}$$ 4. Points (4,3) and (8,6) $$m = \frac{6 - 3}{8 - 4} = \frac{3}{4}$$ $$y - 3 = \frac{3}{4}(x - 4)$$ $$y = \frac{3}{4}x - 3 + 3 = \frac{3}{4}x$$ 5. Points (3,2) and (6,4) $$m = \frac{4 - 2}{6 - 3} = \frac{2}{3}$$ $$y - 2 = \frac{2}{3}(x - 3)$$ $$y = \frac{2}{3}x - 2 + 2 = \frac{2}{3}x$$ All equations are simplified to slope-intercept form.