Chapter 2 - Polynomials
| Thursday, 26 August 2021 | |
| 8-minute read | |
| 1547 words | |
2.1 Polynomials
A polynomial is an equation with multiple terms. Exact definitions may very, but multiple terms may be either 4+ terms, or 1+ terms. It is mostly agreed upon that a polynomial is an umbrella term that also captures monomials, binomials, and trinomials.
All polynomials can not have negative or fraction exponents.
\[ ax^n + ... + c \]
Quadratic Functions
A quadratic function is a function whose highest degree is the second power.
The form \(f(x) = a(x - h)^2 + k\) is known as the standard form of a quadratic. It is also known as the vertex form because of the information about the vertex it conveys to you. It is most similar to the point slope form of a linear equation \(y - y_1 = m(x - x_1)\).
Where \((x_1, y_1)\) is the point a linear equation would intercept, \((h, k)\) is the point of the vertex of a quadratic function.
Examples
- The height of a golf ball as a function of time is modeled by the equation \(h(t) = -16t^2 + 85t + 20\)
In this problem, we have to find the vertex.
\[ x = - \frac{b}{2a} = - \frac{85}{2(-16)} = \frac{85}{32} s \] \[ y = h(x) = 132.891 ft \]
2.2 Graphs of Polynomial Functions
Polynomials will never have in their graphs:
- Cusps
- Discontinuities
- Corners
All non-linear polynomials will form smooth curves.
End Behavior of a Polynomial Function
The end behavior describes what happens at the left and right ends of a function.
One correct format for indicating end behavior is:
\[ \textrm{as } x \to \infty, y \to ? \] \[ \textrm{as } x \to - \infty, y \to ? \]
Other forms may use limit notation.
\[ \lim_{x \to \infty} y \to ? \] \[ \lim_{x \to - \infty} y \to ? \]
Examples
State the end behavior of each function
- \(y = -x^3 + 4x\)
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\[ \lim_{x \to - \infty}{y \to \infty} \]
\[ \lim_{x \to \infty}{y \to - \infty} \]
- \(y = x^4 - 5x^2 + 4\)
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\[ \lim_{x \to - \infty}{y \to \infty} \]
\[ \lim_{x \to \infty}{y \to \infty} \]
- \(y = x^5 - x\)
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\[ \lim_{x \to - \infty}{y \to - \infty} \]
\[ \lim_{x \to \infty}{y \to \infty} \]
Facts about the polynomial \(p(x) + ax^n + bx^{n - 1} ...\)
- \(p(x)\) has at most \(n\) real zeros.
- The graph of \(p(x)\) has at most \(n - 1\) local extrema.
Zeroes
Zeroes and associated items go by four different names.
Given that \(f(x)\) is a polynomial and \(a\) is a real number:
- \(x = a\) is a zero off
- \(x = a\) is a solution to the equation \(f(x) = 0\)
- \((x - a)\) is a factor off
- \((a, 0)\) is an x-intercept of the graph of \(f(x)\)
Examples
- Find the zeroes of the function \(f(x) = x^3 - x^2 - 2x\)
| \(f(x) = x(x^2 - x - 2)\) |
| \(f(x) = x(x + 1)(x - 2)\) |
| \(x(x + 1)(x - 2) = 0\) |
The zeroes are \(\{ -1, 0, 2 \}\)
- Experiment with the exponents \(m\) and \(n\) in the function \(f(x) = (x + 2)^m (x - 3)^n\). Both \(m\) and \(n\) must be positive integers.
The numbers \(m\) and \(n\) give us multiplicity.
An even multiplicity causes the polynomial at that point to bounce off the x-axis. An odd multiplicity causes the polynomial to pass through.
Manually graphing \(p(x)\)
- Find the end behavior
- Find zeroes — by factoring or other methods
- Plot a few intermediate points — find the different points between the zeroes
- Draw the graph
Intermediate Value Theorem (IVT)
The table below gives values of a polynomial function \(p(x)\). Give three one-unit long intervals that contain zeroes of \(p(x)\).
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| \(p(x)\) | -4 | -2 | 1 | -7 | -1 | 6 | 3 |
Give three one-unit long intervals that contain zeroes of \(p(x)\).
\((1, 2), (2, 3) (4, 5)\)
Intermediate Value Theorem (IVT) - any interval in a continuous function between \([a, b]\) is guaranteed to contain the range of \([f(a), f(b)]\) (other y-values may also exist in this interval)
Using the intermediate value theorem, we can determine a sign change in the interval \([m, n]\) will always contain a zero as it guarantees to contain the range passing through 0.
Examples
- Find the expanded form of a polynomial function \(p(x)\) whose zeroes are \(-2, 1, 3\)
\[ (x + 2)(x - 1)(x - 3) \]
\[ (x^2 + x - 2)(x - 3) \]
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| x^2 | x | -2 | |
| x | x^3 | x^2 | -2x |
| -3 | -3x^2 | -3x | 6 |
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- Find the expanded form of a polynomial function \(p(x)\) whose zeroes are \(4 \pom \sqrt{2}\)
\[ (x + 4 - \sqrt{2})(x + 4 + \sqrt{2}) \]
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| x | 4 | \(-\sqrt{2}\) | |
| x | x^2 | 4x | \(-\sqrt{2}x\) |
| 4 | 4x | 16 | \(-\sqrt{32}\) |
| \(\sqrt{2}\) | \(\sqrt{2}x\) | \(\sqrt{32}\) | -2 |
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\[ p(x) = x^2 + 8x + 14 \]
2.3 Real Zeroes of Polynomial Functions
To solve for zeroes of a polynomial, synthetic division can be used.
Show that \((x - 2)\) are factors of \(2x^4 + 7x^3 - 4x^2 - 27x - 17\)
| 2 | 2 | 7 | -4 | -27 | -18 |
| 4 | 22 | 36 | 18 | ||
| 2 | 11 | 18 | 9 | 0 |
Finding zeroes
The possible rational zeroes in a function can be found with \frac{p}{q}, where \(p\) is all the factors of coefficient of the lowest-degree term, and \(q\) is all the factors of the coefficient of the highest-degree term.
Sometimes this result in a lengthy list of possible rational zeroes. For this reason, it's good to have some tools to help trim down the list. Some helpful items include:
- Descartes's Rule of Signs
- Upper bound theorem
- Lower bound theorem
Descartes's Rule of Signs
Descartes's Rule of Signs will only tell you the types of zeroes.
To find the number of positive real zeroes, find the number of variations in signs as you go from term to term in the polynomial. The number of positive real zeroes will be the number of variations \(v\) minus any positive number \(n\) except for \(v - n \lt 0\)
To find the number of negative real zeroes, find the number of variations of \(f(-x)\) (negate all odd-power terms). The number of negative real zeroes will be the number of variatiosn \(v\) minus any positive number \(n\) except for \(v - n \lt 0\)
Example
Find the number of possible positive and negative real zeroes of the function \(f(x) = 3x^3 - 5x^2 + 6x - 4\)
3 sign changes, 3 or 1 possible positive real zeroes.
\(f(-x) = -3x^3 - 5x^2 - 6x - 4\)
0 sign changes, 0 possible negative real zeroes.
Upper and Lower Bounds
Upper and lower bounds establish a floor and ceiling for all possible zeroes.
Using synthetic division, a value is an upper boundPepega Clap WR Pepega Clap WR when the result has no negative terms.
A value is a lower bound when all the sign alternates.
Example
Is 1 from the factor \((x - 1)\) an upper bound, lower bound, or neither for the polynomial below?
\(6x^3 - 4x^2 + 3x - 2\)
| 1 | 6 | -4 | 3 | 2 |
| 6 | 2 | 5 | ||
| 6 | 2 | 5 | 3 |
No negative terms — an upper bound.
2.5 The Fundamental Theorem of Algebra
Recall that real numbers are a subset of complex numbers.
\(\{ 1, 2, 3, ... \} \subset \{ 1, 2, 3, i, 2i, 3i, ... \}\)
Original statement of The Fundamental Theorem of Algebra
If \(p(x)\) is a polynomial of degree \(n\) where \(n > 0\), \(p(x)\) has at least one zero in the comple number system.
Improved form of The Fundamental Theorem of Algebra
An \(n\)-th degree polynomial has exactly \(n\) complex zeroes.
Factored form of The Fundamental Theorem of Algebra
/A polynomial of degree \(n\) has precisely \(n\) linear factors such that \(f(x) = a(x - z_1)(x - z_2)...\) where \(z\) represents a complex number./
Examples
- Show that \(x^3 + 4x\) has exactly 3 zeroes.
\[x(x^2 + 4) = 0\]
\[x = \{0, ...\}\]
\[x^2 + 4 = 0\]
\[x^2 = -4\]
\[x = \pm \sqrt{-4}\]
\[x = \{ 0, \pm 2i \}\]
Given \(f(x) = x^4 - x^2 - 20\)
Factor \(f(x)\) over the rational numbers
- \(f(x) = (x^2 - 5)(x^2 + 4)\)
Factor \(f(x)\) over the real numbers
- \(f(x) = (x - \sqrt{5})(x + \sqrt{5})(x^2 + 4)\)
Factor \(f(x)\) completely
- \(f(x) = (x - \sqrt{5})(x + \sqrt{5})(x - 2i)(x + 2i)\)
2.6 Rational Functions
A rational function can be expressed such that \(R(x) = \frac{N(x)}{D(x)}\) where \(N(x)\) and \(D(x)\) are polynomials. \(D(x)\) must not be the zero polynomial.
What is the domain of the rational function \(R(x)\)? \(x \| D(x) \ne 0\)
Given $R(x) = \frac{x - c}{(x - a)(x - b)}, we can find - Removable discontinuity when \(a\) or \(b = c\) - Vertical asymptote at \(x = a\) and \(x = b\)
Examples
- Find all horizontal and vertical asymptotes of \(f(x) = \frac{2x^2}{x^2 - 1}\)
\[ \frac{2x^2}{x^2 - 1} = \frac{2x^2}{(x-1)(x+1)} \] Asymptotes at \(x = \pm 1\)
- Find all asymptotes and removable discontinuities of \(f(x) = \frac{x^2+x-2}{x^2-x-6}\)
\[ \frac{(x + 2)(x - 1)}{(x - 3)(x + 2)} \]
Removable discontinuities at \((-2, f(-2))\)
Degrees are the same. Horizontal asymptotes at \(y = \frac{x^2}{x^2} = 1\)
Vertical asymptote at \(x = 3\)