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Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. Free to share, print, make copies and changes. Get yours at Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: If you have any questions or problems please

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The Fundamental Theorem of Algebra Finding Polynomials with Given Zeroes Zeroes of Polynomial Functions with Real Coefficients Rational Coefficients Integer Coefficients and the Rational Zeroes Theorem The Rule of Signs Zeroes of Polynomial Functions and Their Theorems Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems Free to share, print, make copies and changes. Get yours at ect&utm_source=boundless

The fundamental theorem of algebra states that every non-constant single- variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. Equivalently (by definition), the fundamental theorem states that the field of complex numbers is algebraically closed. The fundamental theorem is also stated as follows: every non-zero, single- variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division. The Fundamental Theorem of Algebra Free to share, print, make copies and changes. Get yours at theorems-31/the-fundamental-theorem-of-algebra ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess The Maximum Modulus Principle View on Boundless.com Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems

A polynomial constructed from n roots will have degree n or less. That is to say, if given three roots, then the highest exponential term needed will be x3. Each zero given will end up being one term of the factored polynomial. After finding all the factored terms, simply multiply them together to obtain the whole polynomial. Because a polynomial and a polynomial multiplied by a constant have the came roots, every a polynomial is constructed from given zeroes the general solution includes a constant, shown here as c. Finding Polynomials with Given Zeroes Free to share, print, make copies and changes. Get yours at theorems-31/finding-polynomials-with-given-zeroes ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems

Real numbers include all the rational and irrational numbers. For example: −5,4/3, √2 are all real numbers. Zeroes of Polynomial Functions with Real Coefficients Free to share, print, make copies and changes. Get yours at theorems-31/zeroes-of-polynomial-functions-with-real-coefficients ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems

If given the function, [Equation 1], any value of [Equation 2] that will result in [Equation 3] is a root of the function. For this reason, roots are often referred to as a zero of the function. Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems Free to share, print, make copies and changes. Get yours at theorems-31/zeroes-of-polynomial-functions-with-real-coefficients ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 1 View on Boundless.com Equation 2 View on Boundless.com Equation 3 View on Boundless.com

There are many ways to find the roots of a polynomial. If one is confident factoring out polynomials into their simplest forms, its roots can usually be found by inspection. However, if one is not confident, or it is a tricky polynomial, the quadratic equation can be used. Free to share, print, make copies and changes. Get yours at theorems-31/zeroes-of-polynomial-functions-with-real-coefficients ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. A real number that is not rational is called irrational. Irrational numbers include √2, π, and e. Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations. Rational Coefficients Free to share, print, make copies and changes. Get yours at theorems-31/rational-coefficients ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems

In algebra, the Rational Zeros Theorem (also known as Rational Root Theorem, or Rational Root Test) states a constraint on rational solutions (or roots) of the polynomial equation [Equation 4]with integer coefficients. Integer Coefficients and the Rational Zeroes Theorem Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems Free to share, print, make copies and changes. Get yours at theorems-31/integer-coefficients-and-the-rational-zeroes-theorem ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 4 View on Boundless.com

If a0 and an are non-zero, then each rational solution x, when written as a fraction x = p/q in lowest terms (i.e., the greatest common divisor of p and q is 1), satisfies 1) p is an integer factor of the constant term a0, and 2) q is an integer factor of the leading coefficient an. A proof can be derived by first moving the constants to one side, factoring and multiplying by qn. Then a generalized form of Euclid's lemma states that p divides a0. The proof for q is similar. Integer Coefficients and the Rational Zeroes Theorem Free to share, print, make copies and changes. Get yours at theorems-31/integer-coefficients-and-the-rational-zeroes-theorem ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems

The rule of signs gives us an upper bound number of positive or negative roots of a polynomial. It is not a complete criterion, meaning that it does not tell the exact number of positive or negative roots. The rule states that if the terms of a polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less by a multiple of 2. As a corollary of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1 [f(-x)], or fewer than it by a multiple of 2. The Rule of Signs Free to share, print, make copies and changes. Get yours at theorems-31/the-rule-of-signs ?campaign_content=book_196_section_31&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Polynomial and Rational Functions > Zeroes of Polynomial Functions and Their Theorems

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Key terms coprime Having no positive integer factors, aside from 1, in common with one or more specified other positive integers. Euclid's lemma In number theory, Euclid's lemma (also called Euclid's first theorem) is a lemma that captures one of the fundamental properties of prime numbers. It states that if a prime divides the product of two numbers, it must divide at least one of the factors. For example since 133 × 143 = is divisible by 19, one or both of 133 or 143 must be as well. In fact, 19 × 7 = 133. It is used in the proof of the fundamental theorem of arithmetic. irrational number Any real number that cannot be expressed as a ratio of two integers. multiplicity the number of values for which a given condition holds polynomial an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as a_n x^n + a_{n-1}x^{n-1} a_0 x^0. Importantly, because all exponents are positive, it is impossible to divide by x. quotient The number resulting from the division of one number or expression by another. real number An element of the set of real numbers. The set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers. root the number which,when plugged into the equation, will produce a zero. sign positive or negative polarity. zero Also known as a root, a zero is an x value at which the function of x is equal to 0. Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions

Interactive Graph: Changing a Constant A graph of a polynomial with the quadratic equation y=x2+11x+18 and y=2(x2+11x+18). Notice how the intercepts do not change, even when we multiply the function by a constant. Free to share, print, make copies and changes. Get yours at Boundless. "Interactive Graph: Changing a Constant." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Polynomial and Rational Functions

Interactive Graph: Plots of Quadratic Equations Graph of quadratic equations, changing either a, b, or c in the typical equation. Compare all the graphs to the red function,. Varying each coefficient separately include (blue), (green), and (purple). Free to share, print, make copies and changes. Get yours at Boundless. "Interactive Graph: Plots of Quadratic Equations." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Polynomial and Rational Functions

Interactive Graph: Rule of Signs Graph of. We can graphically see there are two solutions to this polynomial. This still fits with the rule of signs, as -1 is a negative root twice in the equation. Free to share, print, make copies and changes. Get yours at Boundless. "Interactive Graph: Rule of Signs." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Polynomial and Rational Functions

Interactive Graph: Multiplying Fractions Graph of a polynomial with the quadratic equation of. We can graph this equation, and in doing so see where it intercepts the y axis, as a means of checking our solutions to this problem. Free to share, print, make copies and changes. Get yours at Boundless. "Interactive Graph: Multiplying Fractions." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Polynomial and Rational Functions

Interactive Graph: Finding Rational Solutions Graph of. One can also use the Rational Zeros Theorem to narrow down the candidates for solutions, then look to see which one is represented by the graphical form of the equation. Free to share, print, make copies and changes. Get yours at Boundless. "Interactive Graph: Finding Rational Solutions." CC BY-SA View on Boundless.comCC BY-SA 3.0https:// on Boundless.com Polynomial and Rational Functions

The Maximum Modulus Principle A plot of the modulus of cos(z) (in red) for z in the unit disk centered at the origin (shown in blue). As predicted by the fundamental theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge). Free to share, print, make copies and changes. Get yours at Wikipedia. "Maximum modulus principle." Public domain View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/Maximum_modulus_principleView on Boundless.com Polynomial and Rational Functions

Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions The fundamental theorem of algebra states that: A) all non-constant single-variable polynomial with complex coefficients has at least 2 complex roots. B) all non-constant double-variable polynomial with complex coefficients has at least one complex root. C) every constant single-variable polynomial with complex coefficients has at least two complex roots. D) all non-constant single-variable polynomial with complex coefficients has at least one complex root.

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// Polynomial and Rational Functions The fundamental theorem of algebra states that: A) all non-constant single-variable polynomial with complex coefficients has at least 2 complex roots. B) all non-constant double-variable polynomial with complex coefficients has at least one complex root. C) every constant single-variable polynomial with complex coefficients has at least two complex roots. D) all non-constant single-variable polynomial with complex coefficients has at least one complex root.

Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions Which of the following represents a polynomial with zeros x=3 and x=-4? A) 2x2+x-12 B) x2+x+12 C) 2x2+2x-24 D) x2+2x-12

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// Polynomial and Rational Functions Which of the following represents a polynomial with zeros x=3 and x=-4? A) 2x2+x-12 B) x2+x+12 C) 2x2+2x-24 D) x2+2x-12

Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions Which of the following polynomials does NOT have zeros x=1 and x=6? A) x2-7x+6 B) 2x2-14x+12 C) 3x2-21x+24 D) 5x2-35x+30

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// Polynomial and Rational Functions Which of the following polynomials does NOT have zeros x=1 and x=6? A) x2-7x+6 B) 2x2-14x+12 C) 3x2-21x+24 D) 5x2-35x+30

Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions Find the zeros of the following function: y=2x2+4x-30. A) x=-3 and x=5 B) x=6 and x=-10 C) x=3 and x=-5 D) x=-6 and x=10

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// Polynomial and Rational Functions Find the zeros of the following function: y=2x2+4x-30. A) x=-3 and x=5 B) x=6 and x=-10 C) x=3 and x=-5 D) x=-6 and x=10

Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions A polynomial with degree five can have at most how many zeros? A) 4 B) 5 C) 0 D) 6

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Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions Which of the following lists all possible rational roots of x4+2x3- 7x2-8x+12? A) ±1, ±12 B) ±1, ±2, ±3, ±4, ±6, ±12 C) ±2, ±3, ±4, ±6 D) ±1/2, ±6/5, ±12/11

Free to share, print, make copies and changes. Get yours at Boundless - LO. "Boundless." CC BY-SA BY-SA 3.0http:// Polynomial and Rational Functions Which of the following lists all possible rational roots of x4+2x3- 7x2-8x+12? A) ±1, ±12 B) ±1, ±2, ±3, ±4, ±6, ±12 C) ±2, ±3, ±4, ±6 D) ±1/2, ±6/5, ±12/11

Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions Which of the following lists all possible rational roots of 2x3+3x-5? A) ±1, ±1/2, ±5, ±5/2 B) ±1, ±5 C) ±2, ±2/5 D) ±1, ±2, ±5

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Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions What is the maximum number of positive roots of x5+_______x4- 23x_______-27x2+166x-120? A) 4 B) 3 C) 5 D) 2

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Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions What is the maximum number of negative roots of x5+3x4- _______3x3-_______7x_______+166x-120? A) 3 B) 2 C) 4 D) 1

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Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions What is the minimum number of complex roots of the following polynomial: x3+_______x_______-4x+3? A) 2 B) 3 C) 1 D) 0

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Attribution Wikipedia. "Polynomial." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Polynomial Wikipedia. "Real number." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Real_number Wikipedia. "Quadratic equation." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Quadratic_equation Wikipedia. "Complex conjugate root theorem." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Complex_conjugate_root_theorem Wiktionary. "root." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/root Wiktionary. "real number." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/real+number Wikipedia. "Rational number." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Rational_number Wiktionary. "irrational number." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/irrational+number Wiktionary. "quotient." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/quotient Wikipedia. "Descartes' rule of signs." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Descartes'_rule_of_signs Wikipedia. "Descartes' rule of signs." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Descartes'_rule_of_signs Wiktionary. "root." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/root Wiktionary. "sign." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/sign Wiktionary. "polynomial." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/polynomial Boundless Learning. "Boundless." CC BY-SA BY-SA 3.0http:// Connexions. "Factoring." CC BY BY 3.0http://cnx.org/content/m18227/latest/?collection=col10624 Wikipedia. "Fundamental theorem of algebra." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions

Wikipedia. "Maximum modulus principle." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Maximum_modulus_principle Wiktionary. "multiplicity." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/multiplicity Wikipedia. "Rational root theorem." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Rational_root_theorem Wiktionary. "coprime." CC BY-SA BY-SA 3.0http://en.wiktionary.org/wiki/coprime Wikipedia. "Euclid's lemma." CC BY-SA BY-SA 3.0http://en.wikipedia.org/wiki/Euclid's%20lemma Free to share, print, make copies and changes. Get yours at Polynomial and Rational Functions