Question

This question asks for the degree, leading coefficient, zeroes, factors, factors with multiplicity, and the final answer

Answer 1

`\textsf{Degree:}\quad 6`

`\textsf{Leading\;coefficient:}\quad (1)/(3456)`

`\textsf{Zeroes:}\quad -6, 2, 5, 8`

`\textsf{Factors:}\quad (x + 6), (x - 2), (x - 5), (x - 8)`

`\textsf{Factors\;with\;multiplicity:}\quad (x + 6)^3(x - 2)(x - 5)(x - 8)`

`\textsf{Equation\;of\;function:}\quad f(x)=(1)/(3456)(x+6)^3(x-2)(x-5)(x-8)`

Explanation:

Zeroes

The zeroes are the x-values of the points where the function crosses the x-axis, so the x-values when f(x) = 0.

Therefore, the zeroes of the given function are:

• x = -6
• x = 2
• x = 5
• x = 8

`\hrulefill`

Factors

According to the factor theorem, if f(x) is a polynomial, and f(a) = 0, then (x - a) is a factor of f(x). Therefore, the factors of the function are the x-values that satisfy f(x) = 0.

Therefore, the factors of the given function are:

• (x + 6)
• (x - 2)
• (x - 5)
• (x - 8)

`\hrulefill`

Multiplicities

The multiplicity of a factor is the number of times the factor appears in the factored form of the equation of the polynomial.

If the behaviour of the x-intercept is like that of a line, i.e. the curve passes directly through the intercept, its multiplicity is one.

Therefore, the factors (x - 2), (x - 5) and (x - 8) have multiplicity one.

The behaviour of the x-intercept at x = -6 is like that of a cubic function (S-shape). There, this zero has multiplicity 3: (x + 6)³.

`\hrulefill`

Degree

So far we have found the zeros, factors and their multiplicities, so we can write a factored form of the function:

`\implies f(x)=a(x+6)^3(x-2)(x-5)(x-8)`

The degree of the function is the highest exponent value of the variables in the polynomial. Therefore, to find the degree of the function, simply sum the exponents of the factors:

`\implies \textsf{Degree}=3 + 1 + 1 + 1 = 6`

Therefore, the degree of the function is 6.

`\hrulefill`

Leading Coefficient

From inspection of the given graph, the y-intercept is (0, -5).

Therefore, to find the leading coefficient (value of a), substitute (0, -5) into the equation and solve for a:

`\begin{aligned}\implies f(0)=a(0+6)^3(0-2)(0-5)(0-8)&=-5\\a(216)(-2)(-5)(-8)&=-5\\-17280a&=-5\\a&=(1)/(3456)\end{aligned}`

Therefore, the leading coefficient of the function is 1/3456.

`\hrulefill`

Equation of the function

Putting everything together, the function in factored form is:

`f(x)=(1)/(3456)(x+6)^3(x-2)(x-5)(x-8)`

In standard form:

`f(x)=(1)/(3456)x^6+(1)/(1152)x^5-(1)/(36)x^4-(37)/(432)x^3+(17)/(24)x^2+(13)/(8)x-5`