Solving Polynomials

A polynomial looks like this:
polynomial example
example of a polynomial

Solving

"Solving" means finding the "roots" ...

A "root" (or "zero") is where the function is equal to zero:

Graph of Inequality
In between the roots the function is either entirely above,
or entirely below, the x-axis

How do you solve polynomials? That depends on the Degree!

Degree

So the first step in solving a polynomial is to find its degree.

The Degree of a Polynomial with one variable is ...

... the largest exponent of that variable.

polynomial

When you know the degree you can also give the polynomial a name!

Degree Name Example Graph Looks Like
0 Constant 7 polynomial degree graphs
1 Linear 4x+3
2 Quadratic x2-3x+2
3 Cubic 2x3-5x2
4 Quartic x4+3x-2 ...
etc ... ... ...

How To Solve

So now you know the degree, how to solve?

  • Read how to solve Linear Polynomials (Degree 1) using simple algebra.
  • Read how to solve Quadratic Polynomials (Degree 2) with a little work,
  • It can be hard to solve Cubic (degree 3) and Quartic (degree 4) equations,
  • And beyond that it can be impossible to solve polynomials directly.

So what do we do with ones we can't solve? Try to solve them a piece at a time!

If you find just one root, you can then reduce the polynomial by one degree, (you will see how later) and this may be enough to solve the whole polynomial.

Here are some main ways to find roots.

1. Basic Algebra

You may be able to solve using simple algebra:

Example: 2x+1

2x+1 is a linear polynomial:

line on a graph

The graph of y = 2x+1 is a straight line

There will be one root, when y=0.

Use Algebra to solve:

Start with: 2x+1 = 0

Subtract 1 from both sides: 2x = -1

Divide both sides by 2: x = -1/2

And that is the solution:

x = -1/2

(You can also see this on the graph)

You can also solve Quadratic Polynomials using basic algebra (read that page for an explanation).

 

2. By experience, or simply guesswork.

It is always a good idea to see if you can do some simple factoring:

Example: x3+2x2-x

This is cubic ... but wait, you can factor out "x":

x3+2x2-x = x(x2+2x-1)

Now we have one root (x=0) and what is left is quadratic, which we can solve exactly.

Or you may notice a pattern you have seen before:

Example: x3-8

Again this is cubic ... but it is also the "difference of two cubes":

x3-8 = x3-23

And so we can turn it into this:

x3-8 = (x-2)(x2+2x+4)

There will be one root at x=2, because:

(2-2)(22+2×2+4) = (0)(22+2×2+4)

And we can also solve the quadratic x2+2x+4 and we are done

 

3. Graphically.

Graph the polynomial and see where it crosses the x-axis.

Function Grapher   You can type the polynomial into the Function Grapher, and then zoom in to find where it crosses the x-axis.

Graphing is a good way to find approximate answers, and you may also get lucky and discover an exact answer.

Caution: before you jump in and graph it, you should really know How Polynomials Behave, so you find all the possible answers!

Factors

This is useful to know! When you factor a polynomial like this:

f(x) = (x-a)(x-b)(x-c)...

Then a, b, c, etc are the roots!

So Linear Factors and Roots are related, know one and you can find the other.

(Read The Factor Theorem for more details.)

Example: f(x) = (x3+2x2)(x-3)

We see "(x-3)", and that means that 3 is a root (or "zero") of the function.

Why?

Well, let us put "3" in place of x:

f(x) = (33+2·32)(3-3)

f(x) = (33+2·32)(0)

Aha! We are multiplying by zero!

So we don't need to do any more calculation as we know the answer will be zero.

How to Check

Now you think you have found a root ... you should check it!

Simply put the root in place of "x": the polynomial should be equal to zero,

Example: 2x3-x2-7x+2

The polynomial is degree 3, and could be difficult to solve. So let us plot it first:

2x^3-x^2-7x+2

The curve crosses the x-axis at three points, and one of them might be at 2. We can check easily, just put "2" in place of "x":

f(2) = 2(2)3-(2)2-7(2)+2 = 16-4-14+2 = 0

Yes! f(2)=0, so we have found a root!

 

How about where it crosses near -1.8:

f(-1.8) = 2(-1.8)3-(-1.8)2-7(-1.8)+2 = -11.664-3.24+12.6+2 = -0.304

No, it isn't equal to zero, so -1.8 will not be a root (but it may be close!)

But we did discover a root, and we can use that to simplify the polynomial, like this

Example: 2x3-x2-7x+2

(continued)

So, f(2)=0 is a root ... that means we also know a factor:

(x-2) must be a factor of 2x3-x2-7x+2

 

Next, divide 2x3-x2-7x+2 by (x-2) using Polynomial Long Division to find:

2x3-x2-7x+2 = (x-2)(2x2+3x-1)

 

So now I can solve 2x2+3x-1 as a Quadratic Equation and I will know all the roots.

That last example showed you how useful it is to find just one root. Just as I promised:

If you find just one root, you can then reduce the polynomial by one degree, and this may be enough to solve the whole polynomial.

How Far Left or Right

When trying to find roots, how far left and right of zero should you go?

There is a way to tell, and there are a few calculations to do, but it is all simple arithmetic. Read Bounds on Zeros for all the details.

Have You Got All The Roots?

There is an easy way to know how many roots there are. The Fundamental Theorem of Algebra says:

Any polynomial of degree n ... has n roots (zeros)
but you may need to use complex numbers

So: number of roots = the degree of polynomial.

Example: 2x3 + 3x - 6

The degree is 3 (because the largest exponent is 3), and so:

There will be 3 roots.

But Some Roots May Be Complex

Yes, indeed, some roots may be complex numbers (ie have an imaginary part), and so will not show up as a simple "crossing of the x-axis" on a graph.

But there is an interesting fact:

Complex Roots always come in pairs!

Complex Conjugate Pairs

So you either get no complex roots, or 2 complex roots, or 4, etc... Never an odd number.

Which means you automatically know this:

Degree Roots Possible Combinations
1 1 1 Real Root
2 2 2 Real Roots, or 2 Complex Roots
3 3 3 Real Roots, or 1 Real and 2 Complex Roots
4 4 4 Real Roots, or 2 Real and 2 Complex Roots, or 4 Complex Roots
etc   etc!

Positive or Negative Roots?

There is also a special way to tell how many of the roots Are negative or positive called the Rule of Signs that you may like to read about.

Multiplicity of a Root

Sometimes a factor appears more than once. We call that Multiplicity:

Multiplicity is how often a certain root is part of the factoring.

Example: f(x) = (x-5)3(x+7)(x-1)2

This could be written out in a more lengthy way like this:

f(x) = (x-5)(x-5)(x-5)(x+7)(x-1)(x-1)

As you can see (x-5) is used 3 times, so the root "5" has a multiplicity of 3, likewise (x+7) appears once and (x-1) appears twice.

So:

  • the root +5 has a multiplicity of 3
  • the root -7 has a multiplicity of 1 (a "simple" root)
  • the root +1 has a multiplicity of 2

Q: Why is this useful?
A: It makes the graph behave in a special way!

When you see a factor like (x-r)n, "n" is the multiplicity, and

  • even multiplicity just touches the axis at "r" (and otherwise stays one side of the x-axis)
  • odd multiplicity crosses the axis at "r" (changes from one side of the x-axis to the other)

You can see this easily in this graph:

Example: f(x) = (x-2)2(x-4)3

(x-2) has even multiplicity, so it just touches the axis at x=2

(x-4) has odd multiplicity, so it crosses the axis at x=4

Like this:

(x-2)^2(x-4)^3

Summary

  • You can directly solve polynomials of Degree 1 (line) and 2 (quadratic)
  • For Degree 3 and up, graphs can be helpful
  • It is also helpful to:
    • Know how far left or right the roots may be
    • Know how many roots (the same as its degree)
    • Estimate how many may be complex, positive or negative
  • Multiplicity is how often a certain root is part of the factoring.

 
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