### Remainder Theorem

**polynomial function**is a function whose rule is a polynomial in one variable, which is a sum or difference of terms of the form $ax^n$, where $a$ is a real number and $n$ is a nonnegative integer. For a polynomial function $f(x)$, the remainder theorem states that when the polynomial is divided by an expression in the form $x - c$, the remainder is $f(c)$. The value of $f(c)$ is the result when the value $c$ is substituted for $x$ in the polynomial. The remainder theorem can be used when the remainder is needed, but the full quotient is not of interest. In that case, completing the full division is unnecessary.

### Factor Theorem

The factor theorem states that if $x - c$ is a factor of a polynomial that defines a function $f(x)$, then $f(c) = 0$. It also states that if $f(c) = 0$, then $x - c$ is a factor of the polynomial that defines $f(x)$. The factor theorem is an application of the fact that when an expression is divided by one of its factors, the remainder is zero.

- If $f(c) = 0$, then the remainder is zero, and $x - c$ is a factor of the polynomial.
- If $f(c)\neq0$, then the remainder is not zero, and $x - c$ is not a factor of the polynomial.

### Rational Zeros Theorem

**zero of a function**is any input value that makes the output of the function equal to zero. The rational zeros theorem states that if $\frac{p}{q}$ is a rational zero of a polynomial function $f(x)$, then $q$ is a factor of the leading coefficient of the polynomial, and $p$ is a factor of the constant term of the polynomial. This means that any rational zero is a fraction formed from the factors of the constant term and leading coefficient of the polynomial that defines the function. All possible rational zeros of a polynomial function can be found by creating fractions with all combinations of the factors of the constant term and leading coefficient. Once the possible zeros are known, they can be tested to determine if they are zeros.

Determine the factors of the constant.

The constant is 3. Its factors are $\pm 1$ and $\pm 3$.

Identify the factors of the leading coefficient.

The leading coefficient is 2. Its factors are $\pm 1$ and $\pm 2$.

### Intermediate Value Theorem

The intermediate value theorem states that if there are two function values that have opposite signs, then there is at least one real zero between their corresponding $x$-values. This means that in order for a function to change from having positive outputs to having negative outputs (or from negative outputs to positive outputs), the graph of the function must cross the $x$-axis.

For example, examine the points of a function: $(-4, 30)$, $(0,-6)$, and $(2,60)$.

- The outputs ($y$-values) of the points $(-4, 30)$ and $(0,-6)$ have opposite signs. So, there must be at least one zero between the $x$-coordinates of the points. Likewise, the outputs of $(0,-6)$ and $(2,60)$ have opposite signs, so there must be at least one zero between the $x$-coordinates of the points.
- There must be at least one zero between –4 and zero, and at least one zero between zero and 2.
- When graphing the function, the graph will show that –3, –2, –1, and 1 are all $x$-intercepts. So, the function has three zeros between –4 and zero, and one zero between zero and 2.

### Fundamental Theorem of Algebra

The fundamental theorem of algebra states that if a polynomial function $f(x)$ has degree $n>0$ and the coefficients are complex, then the function has at least one complex zero. A **complex zero** is a zero in the form $a+bi$, where $a$ and $b$ are real numbers and $i=\sqrt{-1}$. Real zeros can be written as complex zeros of the form $a+bi$, where $b = 0$. If $a+bi$ is a zero of $f(x)$, then its complex conjugate $a-bi$ is also a zero.

The fundamental theorem of algebra leads to several important results. Let $f(x)$ be a polynomial function with degree $n>0$ and leading coefficient $a$ be a real number. Then:

- $f(x)$ has $n$ complex zeros, including multiplicities.
- There are $n$ complex numbers $c_1,c_2,...,c_n$ (that are not necessarily distinct) such that $f(x)=a(x-c_1)(x-c_2)\dots(x-c_n)$.
- The polynomial has $n$ linear factors of the form $(x-c)$, where $c$ is a complex number.
- If a linear factor is repeated, then the corresponding zero has a multiplicity greater than 1.

Complex zeros that are real zeros correspond to $x$-intercepts, but those that are not real do not represent $x$-intercepts.

### Complex Zeros

Polynomial Function with Four $x$-intercepts | Polynomial Function with Two $x$-intercepts |
---|---|

The graph shows a polynomial function of a degree of four. Given $n=4$, the polynomial has four complex zeros and four linear factors. In the function, each linear factor corresponds to an $x$-intercept, which is consistent with the four $x$-intercepts on the graph. |
The graph shows a polynomial function of a degree of four. Given $n=4$, the polynomial has four complex zeros and four linear factors. The graph shows only two $x$-intercepts, so each linear factor does not correspond to an $x$-intercept. Although there are four complex zeros, there are only two real zeros. The zeros that are not real numbers are not represented by $x$-intercepts. |

Polynomial theorems and features of polynomial functions can be used together to graph a polynomial function.

To make a sketch of a polynomial function:

1. Apply the rational zeros theorem to identify the possible rational zeros.

2. Test the possible zeros to determine which are actual zeros of the function.

3. Determine the number of complex zeros, and identify zeros with multiplicity greater than 1.

4. Find additional points on graph, including the $y$-intercept. If the function is even or odd, use symmetry to locate additional points.

5. Determine the end behavior of the function.

A graphing utility, such as a graphing calculator, is a useful tool for graphing complicated polynomial functions.

Use the rational zeros theorem to determine the possible rational zeros.

The constant term is –12. Factors of the constant term are $\pm1$, $\pm2$, $\pm3$, $\pm4$, $\pm6$, $\pm12$.

The leading coefficient is –1. Factors of the leading coefficient are $\pm1$.

The possible zeros are: $\pm1$, $\pm2$, $\pm3$, $\pm4$, $\pm6$, $\pm12$.

Test the possible zeros to determine which are zeros of the function.

Substitute each possible zero in the function.The graph has $x$-intercepts at $x=-2$, $x=1$, and $x=3$.

Determine the number of complex zeros.

The degree of the polynomial is four, so there are four complex zeros including multiplicities. Three zeros have been identified.

The linear factors that represent the three zeros in Step 3 are:Since –2 was already identified as a zero, it has multiplicity 2.

Identify the additional points on the graph, including the $y$-intercept. Use points that result from testing the possible rational zeros in Step 2.

$x$ | $y$ |
---|---|

–2 | 0 |

–1 | –8 |

0 | –12 |

1 | 0 |

2 | 16 |

3 | 0 |

Determine the end behavior of the function.

The degree is four, an even number. So, the graph has the same end behavior as $x$ approaches both positive and negative infinity.

Since the coefficient of the first term, $-x^{4}$, of the polynomial is –1, there are two things to consider:

- As $x$ approaches $-\infty$, $f(x)$ approaches $-\infty$.
- As $x$ approaches $\infty$, $f(x)$approaches $-\infty$.

Determine whether the function is even, odd, or neither.

The function has only terms with an even degree. So, it is even. This means that the graph is symmetric about the $y$-axis.

Use the rational zeros theorem to determine the possible rational zeros.

The constant is 4. Its factors are $\pm 1$, $\pm 2$, and $\pm 4$.

The leading coefficient is –1. Its factors are $\pm1$.

Write all possible fractions using factors of the constant in the numerator and factors of the leading coefficient in the denominator.Test the possible zeros to determine which are zeros of the function. If the function equals zero when the value is substituted for $x$, the value is a zero of the function.

Since the function is even, it is symmetric about the $y$-axis. So, $f(-x)=f(x)$. Test only the positive possible rational zeros.The graph has $x$-intercepts at $x=-2$ and $x=2$.

Determine the number of complex zeros.

The degree of the polynomial is four, so there are four complex zeros, including multiplicities. Two zeros have been identified.

The linear factors that represent the two zeros from Step 2 are:Identify additional points on the graph, including the $y$-intercept. Use points that result from testing the possible rational zeros in Step 2.

$x$ | $y$ |

0 | 4 |

1 | 6 |

2 | 0 |

3 | –50 |

Since the function is even and is symmetric about the $y$-axis, the points $(-1, 6)$, $(-2,0)$, and $(-3, -50)$ are also on the graph.

The value –50 may not show on the graph, depending on the range chosen for the $y$-axis, but it helps to indicate the shape of the graph.

Determine the end behavior of the function.

The degree is 4, an even number. So, the graph has the same end behavior as $x$ approaches both positive and negative infinity.

Since the coefficient of the first term, $-x^{4}$, is –1, there are two things to consider:

- As $x$ approaches $-\infty$, $f(x)$ approaches $-\infty$.
- As $x$ approaches $\infty$, $f(x)$ approaches $-\infty$.