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Describe the graph of the function f(x) = x3 − 18x2 + 107x − 210. Include the y-intercept, x-intercepts, and the shape of the graph.

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Panoyin

f(x) = x³ - 18x² + 107x - 210

X - Intercept: f(x) = x³ - 18x² + 107x - 210
                         0 = x³ - 18x³ + 107x - 210
                         0 = x³ - 7x² - 11x² + 77x + 30x - 210
                         0 = x²(x) - x²(7) - 11x(x) + 11x(7) + 30(x) - 30(7)
                         0 = x²(x - 7) - 11x(x - 7) + 30(x - 7)
                         0 = (x² - 11x + 30)(x - 7)
                         0 = (x² - 6x - 5x + 30)(x - 7)
                         0 = (x(x) - x(6) - 5(x) + 5(6))(x - 7)
                         0 = (x(x - 6) - 5(x - 6))(x - 7)
                         0 = (x - 5)(x - 6)(x - 7)
                         0 = x - 5     or     0 = x - 6     or     0 = x - 7
                      + 5      + 5          + 6      + 6          + 7      + 7
                         5 = x        or       6 = x        or       7 = x
Solution Set: {5, 6, 7}

Y - Intercept: f(x) = x³ - 18x² + 107x - 210
                      f(x) = (0)³ - 18(0)² + 108(0) - 210)
                      f(x) = 0 - 18(0) + 0 - 210
                      f(x) = 0 - 0 - 210
                      f(x) = 0 - 210
                      f(x) = -210

Shape of the Graph: Odd Degree Polynomials With a Positive Leading Coefficient
 

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kadenmartin34

Answer:

X - Intercept: f(x) = x³ - 18x² + 107x - 210

0 = x³ - 18x³ + 107x - 210

0 = x³ - 7x² - 11x² + 77x + 30x - 210

0 = x²(x) - x²(7) - 11x(x) + 11x(7) + 30(x) - 30(7)

0 = x²(x - 7) - 11x(x - 7) + 30(x - 7)

0 = (x² - 11x + 30)(x - 7)

0 = (x² - 6x - 5x + 30)(x - 7)

0 = (x(x) - x(6) - 5(x) + 5(6))(x - 7)

0 = (x(x - 6) - 5(x - 6))(x - 7)

0 = (x - 5)(x - 6)(x - 7)

0 = x - 5 or 0 = x - 6 or 0 = x - 7

+ 5 + 5 + 6 + 6 + 7 + 7

5 = x or 6 = x or 7 = x

Solution Set: {5, 6, 7}

Y - Intercept: f(x) = x³ - 18x² + 107x - 210

f(x) = (0)³ - 18(0)² + 108(0) - 210)

f(x) = 0 - 18(0) + 0 - 210

f(x) = 0 - 0 - 210

f(x) = 0 - 210

f(x) = -210

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