SOLVING POLYNOMIAL EQUATIONS

Objective:  To recall how to solve polynomial equations by factoring.

Recall 1: If a polynomial factors, then it is divisible by any one of  its factors.

For example, the polynomial x^4 + 13x^2 + 36 factors as (x - 2)(x + 2)(x - 3)(x + 3).
This means that any one of the factors divides  x^4 + 13x^2 + 36 without remainder.
Indeed, if you divide   x^4 + 13x^2 + 36 by (x - 2) you will find the result to be  x^3 + 2x^2 - 9x - 18, exactly.

Note 1
Finding the factors of a polynomial equation is not always easy.  The polynomials encountered in algebra textbooks are generally easier to factor than arbitrary ones . The next recall suggests some tricks for finding the factors.

Recall 2: Consider the polynomial

2x^3 - 3x^2 - 5x + 6.

By letting x = 1, we find that the polynomial becomes 0; that  is:

2(1)^3 - 3(1)^2 - 5(1) + 6 = 0.

This means that (x - 1) is a factor of the polynomial.  We   can test this by performing long division.

2x^2 -x - 6
x-1)  2x^3 - 3x^2 - 5x + 6
2x^3 - 2x^2
-x^2 - 5x
-x^2 + x
-6x + 6
-6x + 6
0

From this, we see that

2x^3 - 3x^2 - 5x + 6 = (x - 1)(2x^2 - x - 6).

The quadratic at the right side of this last equation factors   as

2x^2 - x - 6 = (x - 2)(2x + 3),

which means that the original polynomial factors as

2x^3 - 3x^2 - 5x + 6 = (x - 1)(x - 2)(2x + 3)

Recall 3: (Solving polynomial equations by factoring.)

By what was said in Recall 1, the polynomial equation

2x^3 - 3x^2 - 5x + 6 = 0

may be rewritten as

(x - 1)(x - 2)(2x + 3) = 0.

Whenever a polynomial is fully factored into a product of   degree one polynomials, the solutions to the polynomial   equation can be read from the factors.   This comes from   the fact that if a product equals zero, then each factor of the   product must also be zero.  For the case of the polynomial   above, it means that

x - 1 = 0, x - 2 = 0 and  2x + 3 = 0.

We therefore read the solutions from these last equations   as:

x = 1,  x = 2 and x = -2/3 .

Examples

Example 1. Find the solutions to the equation

x^3 - 3x^2 - 10x + 24 = 0.

Solution:   Try the factors of 24.  They are

1, - 1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 24, -24

x = 1 or -1 is not a solution; but x = 2 is a solution, since

(2)^3 - 3(2)^2 - 10(2) + 24 = 0.

If x = 2 is a solution, then x - 2 is a factor of the polynomial

x^3 - 3x^2 - 10x + 24

and so we may use long division to find the other factors.

x^2  -  x  -  12
x - 2) x^3 - 3x^2 - 10x + 24
x^3 - 2x^2
-x^2 - 10x
-x^2 +  2x
-12x + 24
-12x + 24
0

Our polynomial factors as  x^3 - 3x^2 - 10x + 24 = (x - 2)(x^2 - x - 12).
Next, notice that the quadratic on the right side of the above equation factors as x^2 - x - 12 = (x - 4)(x + 3); so,

x^3 - 3x^2 - 10x + 24 = (x - 2)(x - 4)(x + 3).

We may now read the solutions from the factors above; they are:

x = 2, x = 4 and x = -3.

Example 2. Find all solutions to the equation

6x^3 + 14x + 17x^2 + 3 = 0.

Solution: First, rearrange the terms so that they are in descending order of the powers of x.  That means rewrite the equation as

6x^3 + 17x^2 + 14x + 3 = 0.

Next, test the quotients of the factors of 6 and 3; for example, test

1, -1, 3, -3, 1/3, -1/3, 3/6, -3/6, 3/2, -3/2  etc.

In this case, we will not have to go far with the testing, because x = -1 is a solution.  We see that

6(-1)^3 + 17(-1)^2 + 14(-1) + 3. = 0.

We use long division to find the other factors.

6x^2 + 11x + 3
x + 1) 6x^3 + 17x^2 + 14x + 3
6x^3 + 6x^2
11x^2  + 14x
11x^2  + 11x
3x + 3
3x + 3
0

This tells us that the polynomial 6x^3 + 17x^2 + 14x + 3 factors as

(x + 1)(6x^2 + 11x + 3).

The quadratic expression on the right in-turn factors as

(2x + 3)(3x + 1),

which means that the polynomial equation becomes

(x + 1)(2x + 3)(3x + 1) = 0.

We may read the solutions from the factors above; they are:

x = -1,  x = -2/3  and  x = -1/3.

Note
All the solutions are among the quotients of factors that we listed at the beginning,

1, -1, 3, -3, 1/3, -1/3, 3/6, -3/6, 3/2, -3/2  etc.

Terminology:

A value for a variable that makes a polynomial zero is  called a root or zero of the polynomial.

Note
Not every polynomial can be factored as easily as the ones that we have just encountered.  The Fundamental Theorem of Algebra says that any polynomial can be factored into a product of first degree or second degree polynomials.  The second degree polynomials may not have real numbers for roots.  For example x^2 + x + 1 does not have a real number as a root.  Rest assured that we shall not encounter such a case in this Workout.

Example 3. Find all solutions to the equation

x^3 - 2x^2 + 1 = 0.

Solution: Here again, x = 1 is a solution; so, we may use long division to find the other factors of the polynomial x^3 - 2x^2 + 1.

We use long division to find the other factors.

x2 -   x            - 1
x - 1) x^3 - 2x^2 + 0x + 1
x^3 -  x^2
-x^2 + 0x
-x^2 + 1x
-x + 1
-x + 1
0

This tells us that the polynomial x^3 - 2x^2 + 1 factors as (x - 1)(x^2 - x - 1).
Our task is now to factor the quadratic polynomial  x^2 - x - 1.  But this has no obvious factors.  We must use the quadratic formula to get the solutions to
x^2 - x - 1. = 0.

The solutions to this last equation will also be the remaining solutions to the original equation.   From the quadratic formula, we have

x =(1 + sqrt(5))/2 or (1 - sqrt(5))/2

as the solutions of the quadratic.  Therefore, the solutions for the original equation are
x =(1 + sqrt(5))/2 or (1 - sqrt(5))/2   and x = 1.

Note
The real solutions to a polynomial equation is the set of real number values that satisfy the equation.  For example, the set of real solutions to the equation x^3 - 8 = 0 is x = 2, because the factors of x^3 - 8 are x - 2 and x^2 + 2x + 4.  Note that the quadratic factor does not have any real roots, according to the quadratic formula.

EXERCISES

In exercises 1 through 11, find the real solutions to the polynomial equations.

1.   x^2 - x - 2=0

2.    x^4 + 2x^3 + x^2=0

3.      x^3 - x^2 - x + 1=0

4.     3x^3 + 5x^2 + 2x=0

5.     16x^4 - 1=0

6.     x^5 - 2x^4 - x^3 + 2x^2 + x - 2=0

7.     x^4 - x^3 + 5x^2 - 5x=0

8.     8x^4 + 10x^2 - 3=0

9.    x^4 - 2x^3 + 2x^2 - 2x + 1=0

10.     x^4 + 13x^2 + 36=0

11.     4x^4 - 25x^2 + 36=0