The Rational Root Theorem tells us the possible fully simplified rational zeros of a polynomial function with integer coefficients.
If pq is a solution of a_nx^n+... a_1+ a_0=0, then p is a factor of the constant term a_0.
If pq is a solution of a_nx^n+... a_1+a_0=0, then q is a factor of the leading coefficient a_n.
We first must identify the leading coefficient and the constant term in our polynomial.
4x^3-49x+( -60)
Let's list all possible values of the numerator and denominator of a rational zero.
Description
Values
The possible values of the numerator are the factors of the constant term -60.
The possible values of the denominator are the factors of the leading coefficient 4.
± 1, ± 2, ± 4
Now we can list all possible rational zeros.
Denominator
±1
±2
±4
Numerator
± 1
±1
±1/2
±1/4
± 2
±2
±1
±1/2
± 3
±3
±3/2
±3/4
± 4
±4
±2
±1
± 5
±5
±5/2
±5/4
± 6
±6
±3
±3/2
± 10
±10
±5
±5/2
± 12
±12
±6
±3
± 15
±15
±15/2
±15/4
± 20
±20
±10
±5
± 30
±30
±15
±15/2
± 60
±60
±30
±15
Let's compare these values to the x-intercepts of the graph.
We can see that there are three reasonable possibilities.
x=- 5/2, x=- 3/2, x=4
Let's use synthetic division to check x=4 first.
rl IR-0.15cm r 4 & |rr 4& 0& -49& -60 & 16& 64& 60 & c 4& 16 & 15 & 0
We can see that x=4 is indeed a zero, and we also get the factor form of the polynomial.
4x^3-49x-60=(x-4)(4x^2+16x+15)
To find the other zeros, we set the quadratic factor equal to 0 and use the Quadratic Formula to solve the equation. For this, we need the coefficients of the quadratic.
ax^2+bx+c=4x^2+16x+15
⇓
a=4, b=16, c=15
