Algebra Know-It-ALL

518 Review Questions and Answers

When we subtract the “b-constant” from both sides in each of these equations and then divide

through by the “a-coefficient” (which is okay because we know that a 1 ,a 2 , and a 3 are all non-

zero), we get these roots:

x=−b 1 /a 1 or x=−b 2 /a 2 or x=−b 3 /a 3

The real solution set is therefore

X= {−b 1 /a 1 ,−b 2 /a 2 ,−b 3 /a 3 }

Question 25-8

It is possible for two, or even all three, of the roots in Answer 25-7 to be the same? If so, give

examples. If not, explain why not.

Answer 25-8

Yes, this can happen. If two of the roots are identical, then the cubic equation has a total of

two real roots, one of them of multiplicity 2. Here’s an example:

(x− 3)(x+ 2)(− 2 x− 4) = 0

In this equation, the root for each binomial is the value of x that makes the binomial equal to 0.

In order from left to right, those values are

x= 3 or x−2 or x=− 2

If all three of the roots are identical, then the cubic equation has one real root with multiplic-

ity 3. Consider this:

(x− 5)(2x− 10)(− 3 x− 15) = 0

Here, as before, the root for each binomial is the x-value that makes it 0. The roots for each

binomial, in order from left to right, are

x= 5 or x= 5 or x= 5

Question 25-9

What is the binomial factor rule?

Answer 25-9

Imagine that we come across a cubic equation and we put it into the polynomial standard

form, like this:

ax^3 +bx^2 +cx+d= 0

where a,b,c, and d are real numbers, and a≠ 0. The binomial factor rule tells us that a real

numberk is a root of this equation if and only if (x−k) is a factor of the polynomial.