Given:
A graph of a polynomial passes through the point (-1, 60) and has three x-intercepts: (-4,0), (1, 0), and (3, 0).
To find:
The equation of the polynomial.
Solution:
A polynomial is defined as:
[tex]P(x)=a(x-c_1)(x-c_2)...(x-c_n)[/tex]
Where, a is a constant and [tex]c_1,c_2,...,c_n[/tex] are the zeros of the polynomial.
The given polynomial has three x-intercepts (-4,0), (1, 0), and (3, 0). It means [tex]-4,1,3[/tex] are the zeroes of the given polynomial and [tex](x+4),(x-1),(x-3)[/tex] are the factors of the given polynomial.
So, the equation of the polynomial is:
[tex]P(x)=a(x+4)(x-1)(x-3)[/tex] ...(i)
It passes through the point [tex](-1,60)[/tex].
[tex]60=a(-1+4)(-1-1)(-1-3)[/tex]
[tex]60=a(3)(-2)(-4)[/tex]
[tex]60=24a[/tex]
Divide both sides by 24.
[tex]\dfrac{60}{24}=a[/tex]
[tex]2.5=a[/tex]
Putting [tex]a=2.5[/tex] in (i), we get
[tex]P(x)=2.5(x+4)(x-1)(x-3)[/tex]
Therefore, the equation of the given polynomial is [tex]P(x)=2.5(x+4)(x-1)(x-3)[/tex].
