Question

The graph of a quartic function crosses the x-axis at -4 and -2 and touches it at 3. State the equation for the family.

Answer 1

The equation of the quartic function is f(x) = x⁴ - 18 · x² + 6 · x + 72.

How to find the possible equations for a quartic polynomial that passes through the x-axis at three points

Herein we must construct at least a polynomial that satisfies all conditions described in the statement. According to the fundamental theorem of algebra, quartic functions may have no real roots, two real roots or four real roots, which means that one of the roots must have a multiplicity of 2.

The root with a multiplicity of 2 is x = 3 and both x = - 4 and x = - 2 have only a multiplicity of 1, then we have the following expression by using the factor form of the definition of polynomials:

f(x) = (x - 3)² · (x + 4) · (x + 2)

Now we expand the expression to get the standard form:

f(x) = (x² - 6 · x + 9) · (x² + 6 · x + 8)

f(x) = x⁴ - 6 · x³ + 9 · x² + 6 · x³ - 36 · x² + 54 · x + 8 · x² - 48 · x + 72

f(x) = x⁴ - 18 · x² + 6 · x + 72

To learn more on quartic functions: https://brainly.com/question/25285042

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