Algebra Know-It-ALL

Part Three 519

Question 25-10

What is the smallest number of real roots that a single-variable cubic equation with real-number
coefficients and a real-number constant can have? What is the largest number of real roots that
such an equation can have?

Answer 25-10

A cubic equation in one variable, with real coefficients and a real constant, can have one real
root, two real roots, or three real roots. There must always be at least one, but there can never
be more than three.

Chapter 26

Question 26-1

What is the polynomial standard form of an nth-degree equation in the variable x, where n is
a natural number larger than 3?

Answer 26-1

The polynomial standard form of such an equation is

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

where a 1 , a 2 ,a 3 , ... an, and b are real numbers. In addition, the leading coefficient, an, must
not be equal to 0.

Question 26-2

In an equation of the form shown in Answer 26-1, what would happen if the coefficient an,
by which xn is multiplied, were equal to 0?

Answer 26-2

If an= 0, we get the equation

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

The term for xn has vanished, leaving us with the polynomial standard form for a single-variable
equation of degree n− 1:

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

Question 26-3

What is the binomial-to-the-nth form of an nth-degree equation in the variable x?