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Chapter 4: Q. 98 (page 210)

Prove the Rational Zeros Theorem.

[Hint: Letpq, where p and q have no common factors except1and -1, be a zero of the polynomial functionf(x)=anxn+an-1xn-1+鈰�+a1x+a0 whose coefficients are all integers. Show that

anpn+an-1pn-1q+鈰�+a1pqn-1+a0qn

Now, because p is a factor of the first n terms of this equation, p must also be a factor of the terma0qn . Since p is not a factor of q (why?), p must be a factor of a0. Similarly, q must be a factoran .]

Short Answer

Expert verified

For a function f(x)=anxn+an-1xn-1+鈰�+a1x+a0where each coefficient is integers and pqis the rational zero then

p is a factor of the constant term a0

q is factor of leading coefficient an

Step by step solution

01

Step 1. Given data

The given function is

f(x)=anxn+an-1xn-1+鈰�+a1x+a0

zero of the function is -1

the common factor of factors of leading coefficient and the constant term is one

02

Step 2. Zero of function

As zero of function is -1

pq=-1p=-qp+q=0

03

Step 3. Proof

Substitutep+q for x in the function

f(p+q)=anpn+an-1pn-1q+鈰�+a1pqn-1+a0qn

p is a factor of n terms so p will be a factor of a0qn

p and q have a common factor of 1only so p is a factor of a0only

Similarly,q is a factor ofan

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