Algebra Know-It-ALL

522 Review Questions and Answers

The real solution set is therefore

X= {−b 1 /a 1 ,−b 2 /a 2 ,−b 3 /a 3 , ··· −bn/an}

Question 26-8

What are the real roots of the following equation? What is the multiplicity of each root? What

is the real solution set X? What is the degree of the equation?

(x+ 4)(2x− 8)^2 (3x)^5 = 0

Answer 26-8

We take each binomial individually, set it equal to 0, and then solve the resulting first-degree

equations:

x+ 4 = 0

2 x− 8 = 0

3 x= 0

These equations resolve to x=−4,x= 4, and x= 0 respectively. Therefore, the real roots of

the original equation are

x=−4 or x= 4 or x= 0

and the real solution set is X= {−4, 4, 0}. The root −4 has multiplicity 1. The root 4 has multi-

plicity 2. The root 0 has multiplicity 5. The degree of the equation is the sum of the exponents

attached to the factors, in this case 1 + 2 + 5, or 8.

Question 26-9

What is the largest number of real roots that a single-variable equation of the nth degree can

have? What is the largest number of real or complex roots that such an equation can have?

Answer 26-9

A single-variable equation of the nth degree can have, at most, n roots in total, considering the

real-number roots and the complex-number roots combined.

Question 26-10

There’s a way to find all the rational-number roots of an nth-degree equation in the single vari-

ablex when it appears in the polynomial standard form

anxn+an− 1 xn−^1 +an− 2 xn−^2 + ··· +a 1 x+b= 0

where a 1 ,a 2 ,a 3 , ... an, and b are nonzero rationals, and n is a positive integer greater than 3.

What is that process?