Question

Find a polynomial function with the zeros -2​, 3​, 4 whose graph passes through the point (6,144)

Answer 1

Answer:
f(x) = 3x^3 - 9x^2 - 2x + 36.

Step by step loved:
If a polynomial function has zeros at -2, 3, and 4, then it can be expressed in factored form as:

f(x) = a(x + 2)(x - 3)(x - 4)

where a is a constant coefficient.

To determine the value of a, we can use the point (6, 144) that the graph passes through. Substituting x = 6 and y = 144 into the equation above, we get:

144 = a(6 + 2)(6 - 3)(6 - 4)
144 = a(8)(3)(2)
144 = 48a
a = 144/48
a = 3

Therefore, the polynomial function that satisfies the given conditions is:

f(x) = 3(x + 2)(x - 3)(x - 4)

Expanding this polynomial gives:

f(x) = 3x^3 - 9x^2 - 2x + 36

So the function is f(x) = 3x^3 - 9x^2 - 2x + 36.

Answer 2

According to the factor theorem, If r is the root of a polynomial then x - r is the factor of the polynomial.

The function with zeros -2, 3, 4 has a form of:

• f(x) = a(x - (-2))(x - 3)(x - 4) = a(x + 2)(x - 3)(x - 4), where a- coefficient

We know that point (6, 144) is on the graph of the function, then find the value of a by substituting the coordinates:

• 144 = a(6 + 2)(6 - 3)(6 - 4)
• 144 = a(8)(3)(2)
• 144 = 48a
• a = 144/48
• a = 3

The function is:

• f(x) = 3(x + 2)(x - 3)(x - 4), in the factor form

or

• f(x) = 3x³ - 15x² - 6x + 72, in the standard form