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Elenna [48]
2 years ago
13

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.

Mathematics
1 answer:
nekit [7.7K]2 years ago
6 0

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

<h3>How to find the possible equations for a quartic polynomial that passes through the x-axis at three points</h3>

Herein we must construct at least a polynomial that satisfies all conditions described in the statement. According to the fundamental theorem of algebra, <em>quartic</em> functions may have no <em>real</em> roots, two <em>real</em> roots or four <em>real</em> 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 <em>factor</em> form of the definition of polynomials:

f(x) = (x - 3)² · (x + 4) · (x + 2)

Now we expand the expression to get the <em>standard</em> 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: brainly.com/question/25285042

#SPJ1

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