Question

Suppose the graph of a cubic polynomial function has the same zeroes and passes through the coordinate (0, –5). Describe the steps for writing the equation of this cubic polynomial function.

Answer 1

Let x=a be the same zeros for the polynomial.
The function is of the form
y = b(x - a)^3
where b  is a constant.

Because the curve passes through (0, -5), therefore
-5 = b(-a)^3  => b = 5/a^3

The equation for the polynomial is
y = (5/a^3)*(x-a)^3
   = 5(x/a - 1)^3

Answer: y = 5(x/a - 1)^3

Answer 2

Answer:

The cubic polynomial is $y=5(\frac{x}{a}-1)^3$.

Step-by-step explanation:

It is given that the degree of the polynomial is 3 and the polynomial has same roots. The polynomial passing through (0,-5).

The polynomial function is defined as

$P(x)=c(x-a_1)^{m_1}(x-x_2)^{m_2}...(x-x_n)^{m_n}$

Where c is a constant, $a_1,a_2,....,a_n$ are roots with multiplicity $m_1,m_2,...,m_n$.

Since the degree of the polynomial is 3 and it has same roots therefore the multiplicity of root is 3.

$y=c(x-a)^3$

The polynomial passing through (0,-5).

$-5=c(0-a)^3$

$-5=c(-a)^3$

$5=ca^3$

$c=\frac{5}{a^3}$

So the equation of the polynomial is,

$y=\frac{5}{a^3}(x-a)^3$

$y=5(\frac{x-a}{a})^3$

$y=5(\frac{x}{a}-1)^3$

Where a is the root of polynomial.

Therefore the cubic polynomial is $y=5(\frac{x}{a}-1)^3$.