The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra says that “every one-variable polynomial of degree $n$ has exactly $n$ complex roots.”

Complex Roots

The standard form of a complex number is $a + bi$, $a$ being real and $bi$ being imaginary. All real roots are complex roots. This means numbers like $11$, $291$, and $3$ can all be roots. All real numbers are complex numbers, and all real roots are complex roots, because when $b = 0$, the number is equal to $a$.

Degree of 2

Let’s look at this equation

\[ x^2 - 3x + 2 \]

The leading term ($x$ squared) has a degree of 2. If we graph this, we should have exactly 2 complex roots.

The code

Here’s the code behind the graph:

Code

q <- function(x) {
  x^2 - 3*x + 2
}

ggplot() +
  geom_hline(yintercept = 0) +
  geom_vline(xintercept = 0) +
  geom_function(fun = q, color = "red") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  labs(x = "X",
       y = "Y")+
  theme_linedraw()+
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

We can see that the parabola intersects the x-axis at 1 and 2. These are our 2 complex roots! Roots, or zeros, are the intercepts on a graph, or the place where $y$ (or $x$ if the parabola is horizontal) is equal to zero.

Degree of 3

All real roots

Let’s use the equation

\[x^3-4x^2+2x+3\] This is a cubic polynomial (a polynomial with degree $3$), which means it will have exactly 3 roots.

The code

Here’s the code behind the graph:

The roots here all pass through the x-axis, and the various intercepts/zeros are $-0.6$, $1.6$, and $3$. But a cubic polynomial passing through the x-axis all three times isn’t always the case.

Odd degree polynomials

A polynomial with an odd degree must have at least 1 real root. Nonreal roots come in conjugate pairs, which means that there would be an even number

When something is a conjugate pair, you flip the sign of the imaginary term. Subtraction becomes addition, addition becomes subtraction.

$a+bi$ is the conjugate of $a - bi$

We can write conjugate pairs like $a\pm bi$

Nonreal Roots with odd degree polynomials

\[ 2x^3 - 5x^2 + 3x +4 \] This equation is a cubic polynomial. Let’s graph this:

The code

Here’s the code behind the graph:

The cubic polynomial only passes through the x-axis one time. The “down-up” part of the graph has no intercepts. These two roots are nonreal. The roots for this equation are $0.5$ and $\approx1.5 \pm 0.97i$

Important

Nonreal roots ALWAYS come in conjugate pairs

We can draw a different line for the x-intercepts that are higher up. This shows that they are roots, they just don’t pass through the x-axis.

The code

Here’s the code behind the graph:

Code

t <- function(x) {
  2*x^3 - 5*x^2 + 3*x +4
}

ggplot() +
  geom_hline(yintercept = 0) +
  geom_hline(yintercept = 4.2, alpha = 0.4, color = "blue")+
  geom_vline(xintercept = 0) +
  geom_function(fun = t, color = "red") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  labs(x = "X",
       y = "Y")+
  theme_linedraw()+
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

Even degree polynomials

Real roots with even degree polynomials

Parabolas—being to the 2nd power, can only have 2 roots. This means that both of them are real, or both of them are nonreal. Quartic polynomials are a little bit different. They can either have all real, 2 real, or none real. Let’s take a look at this quartic polynomial.

\[x^4-2x^3-3x^2+5x+1\]

The roots here are $-1.6$, $-0.2$, $0.5$, and $2.3$. We have exactly 4, and all of them are real.

Nonreal roots with even degree polynomials

All nonreal

\[x^4-1x^3-3x^2+3x+7\] This polynomial has degree 4, so it will have 4 roots.

As you can see, no lines intersect the x-axis, and none of the roots are real. The roots here are $\approx-1.3\pm0.6i$ and $\approx1.7\pm0.9i$

Some nonreal

We can go through the same process for a graph that produces 2 real and 2 nonreal roots

\[x^4-x^3-4x^2+x-3\]

The roots here are $-2$, $2.5$, with the conjugate pair $\approx 0.1\pm0.7i$

Multiplicity

Multiplicity occurs when a line “bounces” off of the x (or y) axis. Let me show you what I mean…

In this graph, the first “down-up” doesn’t technically intersect the x axis, but it’s also not floating around elsewhere. This is a double root. Double roots are pretty much the same things as multiplicity.

Double Roots

When we find the roots of any polynomial, we are simplifying it. Here’s an example of a quartic polynomial equation with a double root (not simplified)

\[f(x) = x^4+3x^3-17x^2-39x-20\] We can start factoring and simplifying to find the roots

Synthetic division

This is not a super in-depth tutorial on how to do synthetic division, but hopefully it’s somewhat helpful. Synthetic division is a really cool trick that uses the coefficients of a polynomial to divide and factor. The coefficients of this polynomial are $1$, $3$, $-17$, $-39$, and $-20$. You line up your coefficients in the order they appear in the equation, like so:

\[\begin{array}{rrrrr} 1 & 3 & -17 & -39 & -20 \\ \end{array}\]

After that, you are then ready to start “dividing”.

$0$, $1$, and $-1$ are always good numbers to start with

\[\begin{array}{c|rrrrr} 0 & 1 & 3 & -17 & -39 & -20 \\ & & 0 & 0 & -13 & 0 \\ \hline & 1 & 3 & -17 & -39 & -20\\ \end{array}\]

You put the starting number in front of all your lined up coefficients and drag down the first coefficient 2 lines. Then, you multiply and add across ($1*0 = 0$, $3 + 0 = 3$, $3*0 = 0$, $-17+0 = -17$, etc.) The very last number is what the “root” is, and we want that number to be $0$. Eventually, continuing with 0 as the starting number, we get $f(0)=-20$, which is not a root. We do the same thing with $1$.

\[\begin{array}{c|rrrrr} 1 & 1 & 3 & -17 & -39 & -20 \\ & & 1 & 4 & -13 & -52 \\ \hline & 1 & 4 & -13 & -52 & -72\\ \end{array}\]

Not a root. What about $-1$?

\[\begin{array}{c|rrrrr} -1 & 1 & 3 & -17 & -39 & -20 \\ & & -1 & -2 & 19 & 20 \\ \hline & 1 & 2 & -19 & -20 & 0\\ \end{array}\]

Huzzah! We have a root! We need 4 though, so we press on. We can’t use the same polynomial that we were using before though, the numbers at the bottom (not including the $0$) make our new polynomial. We start 1 degree lower, so the leading term is to the power of 3. We also need to keep that $-1$ somewhere, which is where the factoring comes in. We can write our new polynomial multiplied by $(x+1)$, so when it’s multiplied out, we keep the same $f(x)$.

Our new polynomial is $(x+1)(x^3 + 2x^2-19x-20)$, so we list the coefficients again:

\[\begin{array}{rrrr} 1 & 2 & -19 & -20 \\ \end{array}\]

And go through the multiplication and addition again:

\[\begin{array}{c|rrrr} -1 & 1 & 2 & -19 & -20 \\ & & -1 & -1 & 20 \\ \hline & 1 & 1 & -20 & 0 \\ \end{array}\]

I started with $-1$ because I already knew that it was a root, in different equations it does require a little bit more trial and error. We have another root! Our new polynomial is $(x+1)(x+1)(x^2+x-20)$. The quadratic factors down into $(x-4)(x+5)$, so our fully factored polynomial is:

\[ f(x) = x^4+3x^3-17x^2-39x-20 = (x+1)^2(x-4)(x+5) \] The double root is $(x+1)$, or $-1$.

This means that all of the roots are $-1$, $-1$, $4$, and $-5$.

Faster way to get the roots

You can do the factoring and simplification by hand with synthetic division. Or you can make your life just a little bit easier and make the computer do it:

Code

k <- function(x) {x^4+3*x^3-17*x^2-39*x-20}

k(1)

[1] -72

Code

k(0)

[1] -20

Code

k(4)

[1] 0

Code

k(-5)

[1] 0

Code

k(-1)

[1] 0

And here’s the graph!

This is a very large quartic polynomial, and goes all the way down to -135, which is why it cuts off in some places. The only place that really matters for our purposes today is the x axis, and you are able to see where the line crosses and bounces off the x axis.

Review

Here are key points about the Fundamental Theorem of Algebra

If the degree is $n$ there will be exactly $n$ roots
All real roots are complex roots, all real numbers are complex numbers
Nonreal roots come in pairs
Polynomials with an odd degree must have at least 1 real root
If the root crosses over or bounces off the x (or y) axis, it is real
If the root is elsewhere, it is nonreal

--- title: "The Fundamental Theorem of Algebra" subtitle: "Polynomial Degrees and Roots" date: "April 2025" categories: [math] image: "zoomed_in_parabola.png" format: html: code-fold: show code-tools: true --- ```{r echo=FALSE, warning = FALSE, message = FALSE} library(tidyverse) ``` ## The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra says that "every one-variable polynomial of degree $n$ has exactly $n$ complex roots." ## Complex Roots The standard form of a complex number is $a + bi$, $a$ being real and $bi$ being imaginary. All real roots are complex roots. This means numbers like $11$, $291$, and $3$ can all be roots. All real numbers are complex numbers, and all real roots are complex roots, because when $b = 0$, the number is equal to $a$. ### Degree of 2 Let's look at this equation $$ x^2 - 3x + 2 $$ The leading term ($x$ squared) has a degree of 2. If we graph this, we should have exactly 2 complex roots. ```{r echo = FALSE, warning = FALSE, message = FALSE} q <- function(x) { x^2 - 3*x + 2 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = q, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` ::: {.callout-note collapse=true icon=false} ## The code Here's the code behind the graph: ```{r warning = FALSE, message = FALSE} q <- function(x) { x^2 - 3*x + 2 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = q, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` ::: We can see that the parabola intersects the x-axis at 1 and 2. These are our 2 complex roots! Roots, or zeros, are the intercepts on a graph, or the place where $y$ (or $x$ if the parabola is horizontal) is equal to zero. ### Degree of 3 #### All real roots Let's use the equation $$x^3-4x^2+2x+3$$ This is a cubic polynomial (a polynomial with degree $3$), which means it will have exactly 3 roots. ```{r echo = FALSE, warning = FALSE, message = FALSE} q <- function(x) { x^3 - 4*x^2 + 2*x +3 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = q, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` ::: {.callout-note collapse=true icon=false} ## The code Here's the code behind the graph: ```{r echo = FALSE, warning = FALSE, message = FALSE} v <- function(x) { x^3 - 4*x^2 + 2*x +3 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = v, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` ::: The roots here all pass through the x-axis, and the various intercepts/zeros are $-0.6$, $1.6$, and $3$. But a cubic polynomial passing through the x-axis all three times isn't always the case. ## Odd degree polynomials A polynomial with an odd degree **must** have at least 1 real root. Nonreal roots come in conjugate pairs, which means that there would be an even number When something is a conjugate pair, you flip the sign of the imaginary term. Subtraction becomes addition, addition becomes subtraction. $a+bi$ is the conjugate of $a - bi$ We can write conjugate pairs like $a\pm bi$ ### Nonreal Roots with odd degree polynomials $$ 2x^3 - 5x^2 + 3x +4 $$ This equation is a cubic polynomial. Let's graph this: ```{r echo = FALSE, warning = FALSE, message = FALSE} f <- function(x) { 2*x^3 - 5*x^2 + 3*x +4 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = f, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` ::: {.callout-note collapse=true icon=false} ## The code Here's the code behind the graph: ```{r echo = FALSE, warning = FALSE, message = FALSE} f <- function(x) { 2*x^3 - 5*x^2 + 3*x +4 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = f, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` ::: The cubic polynomial only passes through the x-axis one time. The "down-up" part of the graph has no intercepts. These two roots are nonreal. The roots for this equation are $0.5$ and $\approx1.5 \pm 0.97i$ ::: {.callout-important} Nonreal roots ALWAYS come in conjugate pairs ::: ```{r echo = FALSE, warning = FALSE, message = FALSE} t <- function(x) { 2*x^3 - 5*x^2 + 3*x +4 } ggplot() + geom_hline(yintercept = 0) + geom_hline(yintercept = 4.2, alpha = 0.4, color = "blue")+ geom_vline(xintercept = 0) + geom_function(fun = t, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` We can draw a different line for the x-intercepts that are higher up. This shows that they are roots, they just don't pass through the x-axis. ::: {.callout-note collapse=true icon=false} ## The code Here's the code behind the graph: ```{r warning = FALSE, message = FALSE} t <- function(x) { 2*x^3 - 5*x^2 + 3*x +4 } ggplot() + geom_hline(yintercept = 0) + geom_hline(yintercept = 4.2, alpha = 0.4, color = "blue")+ geom_vline(xintercept = 0) + geom_function(fun = t, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` ::: ## Even degree polynomials ### Real roots with even degree polynomials Parabolas—being to the 2nd power, can only have 2 roots. This means that both of them are real, or both of them are nonreal. Quartic polynomials are a little bit different. They can either have all real, 2 real, or none real. Let's take a look at this quartic polynomial. $$x^4-2x^3-3x^2+5x+1$$ ```{r echo = FALSE, warning = FALSE, message = FALSE} m <- function(x) { x^4-2*x^3-3*x^2+5*x+1 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = m, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` The roots here are $-1.6$, $-0.2$, $0.5$, and $2.3$. We have exactly 4, and all of them are real. ### Nonreal roots with even degree polynomials #### All nonreal $$x^4-1x^3-3x^2+3x+7$$ This polynomial has degree 4, so it will have 4 roots. ```{r warning = FALSE, message = FALSE, echo = FALSE} g <- function(x) { x^4-1*x^3-3*x^2+3*x+7 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = g, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` As you can see, no lines intersect the x-axis, and none of the roots are real. The roots here are $\approx-1.3\pm0.6i$ and $\approx1.7\pm0.9i$ #### Some nonreal We can go through the same process for a graph that produces 2 real and 2 nonreal roots $$x^4-x^3-4x^2+x-3$$ ```{r warning = FALSE, message = FALSE, echo = FALSE} h <- function(x) { x^4-x^3-4*x^2+x-3 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = h, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` The roots here are $-2$, $2.5$, with the conjugate pair $\approx 0.1\pm0.7i$ ## Multiplicity Multiplicity occurs when a line "bounces" off of the x (or y) axis. Let me show you what I mean... ```{r warning = FALSE, message = FALSE, echo = FALSE} w <- function(x) { x^4-2.2*x^3-3*x^2+2*x+2 } ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = w, color = "red") + coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5) ) ``` In this graph, the first "down-up" doesn't technically intersect the x axis, but it's also not floating around elsewhere. This is a **double root**. Double roots are pretty much the same things as multiplicity. ### Double Roots When we find the roots of any polynomial, we are simplifying it. Here's an example of a quartic polynomial equation with a double root (not simplified) $$f(x) = x^4+3x^3-17x^2-39x-20$$ We can start factoring and simplifying to find the roots ## Synthetic division This is not a super in-depth tutorial on how to do synthetic division, but hopefully it's somewhat helpful. Synthetic division is a *really* cool trick that uses the coefficients of a polynomial to divide and factor. The coefficients of this polynomial are $1$, $3$, $-17$, $-39$, and $-20$. You line up your coefficients in the order they appear in the equation, like so: $$\begin{array}{rrrrr} 1 & 3 & -17 & -39 & -20 \\ \end{array}$$ After that, you are then ready to start "dividing". $0$, $1$, and $-1$ are always good numbers to start with $$\begin{array}{c|rrrrr} 0 & 1 & 3 & -17 & -39 & -20 \\ & & 0 & 0 & -13 & 0 \\ \hline & 1 & 3 & -17 & -39 & -20\\ \end{array}$$ You put the starting number in front of all your lined up coefficients and drag down the first coefficient 2 lines. Then, you multiply and add across ($1*0 = 0$, $3 + 0 = 3$, $3*0 = 0$, $-17+0 = -17$, etc.) The very last number is what the "root" is, and we want that number to be $0$. Eventually, continuing with 0 as the starting number, we get $f(0)=-20$, which is not a root. We do the same thing with $1$. $$\begin{array}{c|rrrrr} 1 & 1 & 3 & -17 & -39 & -20 \\ & & 1 & 4 & -13 & -52 \\ \hline & 1 & 4 & -13 & -52 & -72\\ \end{array}$$ Not a root. What about $-1$? $$\begin{array}{c|rrrrr} -1 & 1 & 3 & -17 & -39 & -20 \\ & & -1 & -2 & 19 & 20 \\ \hline & 1 & 2 & -19 & -20 & 0\\ \end{array}$$ Huzzah! We have a root! We need 4 though, so we press on. We can't use the same polynomial that we were using before though, the numbers at the bottom (not including the $0$) make our new polynomial. We start 1 degree lower, so the leading term is to the power of 3. We also need to keep that $-1$ somewhere, which is where the factoring comes in. We can write our new polynomial multiplied by $(x+1)$, so when it's multiplied out, we keep the same $f(x)$. Our new polynomial is $(x+1)(x^3 + 2x^2-19x-20)$, so we list the coefficients again: $$\begin{array}{rrrr} 1 & 2 & -19 & -20 \\ \end{array}$$ And go through the multiplication and addition again: $$\begin{array}{c|rrrr} -1 & 1 & 2 & -19 & -20 \\ & & -1 & -1 & 20 \\ \hline & 1 & 1 & -20 & 0 \\ \end{array}$$ I started with $-1$ because I already knew that it was a root, in different equations it does require a little bit more trial and error. We have another root! Our new polynomial is $(x+1)(x+1)(x^2+x-20)$. The quadratic factors down into $(x-4)(x+5)$, so our fully factored polynomial is: $$ f(x) = x^4+3x^3-17x^2-39x-20 = (x+1)^2(x-4)(x+5) $$ The double root is $(x+1)$, or $-1$. This means that all of the roots are $-1$, $-1$, $4$, and $-5$. ## Faster way to get the roots You can do the factoring and simplification by hand with synthetic division. Or you can make your life just a little bit easier and make the computer do it: ```{r warning=FALSE, message=FALSE} k <- function(x) {x^4+3*x^3-17*x^2-39*x-20} k(1) k(0) k(4) k(-5) k(-1) ``` And here's the graph! ```{r warning = FALSE, message = FALSE, echo = FALSE} k <- function(x) {x^4+3*x^3-17*x^2-39*x-20} ggplot() + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + geom_function(fun = k, color = "red") + coord_cartesian(xlim = c(-10, 10), ylim= c(-50, 10)) + scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+ scale_y_continuous(breaks = -50:10, limits = c(-50, 10))+ labs(x = "X", y = "Y")+ theme_linedraw()+ theme( panel.grid.minor = element_blank(), axis.title.y = element_text(angle = 0, vjust = 0.5), axis.text.y = element_blank(), axis.ticks.y = element_blank() ) ``` This is a very large quartic polynomial, and goes all the way down to -135, which is why it cuts off in some places. The only place that really matters for our purposes today is the x axis, and you are able to see where the line crosses and bounces off the x axis. ## Review Here are key points about the Fundamental Theorem of Algebra * If the degree is $n$ there will be exactly $n$ roots * All real roots are complex roots, all real numbers are complex numbers * Nonreal roots come in pairs * Polynomials with an odd degree **must** have at least 1 real root * If the root crosses over or bounces off the x (or y) axis, it is real * If the root is elsewhere, it is nonreal