Rational Zero Theorem
posted by Chopsticks .
I know how to solve if the leading coefficient is 1, but how would i solve it if its not 1.
Ex.
f(x) = 2x^3 + 7x^2  7x + 30
Can someone help me start that out? Then I'll use synthetic divison to solve for it.

If someone looks at this, please reply. I will for replies around 6:30am before school starts.

If the rational root is p/q, then p must be a divisor of 30 and q a divisor of 2.

Also, if there are many possible candidates like in this case, you should not try them all out. Instead, you should proceed as follows.
Let's denote the polynomial by P(x).
Suppose y is a possible rational root. But then we check if y is a root and see that P(y) is not equal to zero. You can then apply the Rational Roots Theorem to the polynomial:
Q(x) = P(y + x)
You don't have to actually replace x by y + x in the original polynomial and expand out all the powers. All you need to know to apply the Rational Roots Theorem are the coefficients of the highest power and the constant term.
The coefficient of the highest power of x remains the same, the constant term is P(y).
In this case, if you take y = 1, then
P(y) = 32. So, the possible rational roots of Q(x) are of the form plus or minus 2^n for n ranging from minus 1 to 5. Therefore the possible rational roots of the original polynomial are of the form:
1 plus or minus 2^n
for 1<=n<=5
You then check which of the rational root candidates on your original list are of this form. You are then left with:
x = 1/2, 3/2, 1, 3, 3, 5, 15.
If we check x = 1, we find:
P(1) = 42.
This then generates the following candidate roots:
x = 1 + p/q
with p a divisor of 42 and q a divisor of 2. The remaining candidates are then:
x = 3 and x = 5.
None of them work, so there are no rational roots (provided I didn't make any mistakes :) ) 
Thanks alot!

f(x)=x^3+2x^2+4x+1

give more example by solving rational zero theorem.

x37x2+11x5=0
Respond to this Question
Similar Questions

Algebra
Can someone please explain how to do these problems. 1)write a polynomial function of least degree with intregal coefficients whose zeros include 4 and 2i. 2)list all of the possible rational zeros of f(x)= 3x^32x^2+7x+6. 3)Find all … 
Algebra
Using rational zero theorem find all rational zeros P(x)=3x^3+43x^2+43x+27 Factors: 27, 1, 3, 9, 27 Factors: 3, 1, 3 +1/3, +1, +3, +9, +27 possible rational zero for function P(x) 
math
Hi, I am trying to solve a problem using Chebyshev's Theorem. The problem says that: A large sample of Northern Pike caught at Taltson Lake (Canada) showed that the average length was x (mean)=32.5 inches with sample standard deviation … 
Precalculus
I'm HOPING this is the last problem I need help on.. :/ List the possible rational zeros of f using the rational zero theorem. f(x) = 2x^3  x^2 + 5x + 6 I applied the rational zero theorem, but none of the factors worked :( 
College Algebra
Use rational root theorem and the factor theorem to help solve the following equation X42x313x2+38x24=0 
Pre Calculus
1. Find all rational zeros of the polynomial. Then determine any irrational zeros, and factor the polynomial completely. 3x^411x^3+5x^2+3x 2. Find the polynomial with leading coefficient 1 that has a degree of 4, a zero of multiplicity … 
algebra
Use the rational zero theorem to list all possible rational zeros for the given function f(x)=x^34x^219x14 
College Algebra 105
use the rational zero's theorem to list the potential zero's of the polynomial function. Do not attempt to find the zero's f(x)=169x^7x^6+x+13 and then the second part is find the potential rational zero's. 
Algebra 2
Did I solve this problem correctly? Directions: State the possible rational zeros for each function. Question:f(x) = x6  64 Answer: Constant term:64 Factors:1,2,4,8,16,32,64 Leading coefficient:1 Factors: 1 ±1,2,4,8,16,32,64/1 = 
Algebra
Solve the given polynomial equation. Use the Rational Zero Theorem and Descartes's Rule of Signs as an aid in obtaining the first root. 2 x cubed minus 5 x squared minus 5 x minus 1 equals 02x3−5x2−5x−1=0