Name: eriod: Date: ALGEBRA II Notes for 5.5,5.6 Theorems about Roots of olynomial Equations and The fundamental theorem of Algebra. What you ll learn To solve equations using the Rational Root Theorem. To use the conjugate Root Theorem. To use the Fundamental Theorem of Algebra to solve polynomial equations with comple solutions. Vocabulary: Rational Root Theorem. Conjugate Root Theorem. Descartes Rule of Signs. Fundamental theorem of Algebra. Factoring a polynomial can be challenging, but there is a theorem to help you with that. Rational Root Theorem : n n1 1 0 a a a a Let n n1... 1 0 be a polynomial with integerscoefficients. Thereare limitedof possible roots of 0 p * Rational roots must have reduced form where p isan integer factor of a0 and q isan integer factor of q a n roblem 1: Finding a Rational Root 5 0 What are the rational roots of? Leading coefficient (±1,±) and constant term 5 (±1, ±5) The only possible rational roots have the form The only possible roots are 1-1 5-5 1/ -1/ 5/ -5/ () 8 0 0-80 6 7/ 5-75/ the only rational root of 5 0 is1 Note: the Rational Root Theorem does not necessarily give the zeros of the equation. It provides a list of first guesses to test as roots. -5-10 0? What are the rational roots of? Once you find one root, use synthetic division to factor the polynomial. Continue finding roots and dividing until you have a second degree polynomial. Use the Quadratic Formula to find the remaining roots.
roblem : Using the Rational Root Theorem What are the rational roots of 15 0? Leading coefficient 1 (±1,±,±5,±15) and constant term (±1, ±) The only possible rational roots have the form Test each possible rational root in 15 0 until you find a root 1 1 1 1,,,,,,, 5 5 15 15 Remember steps to find rational roots. 1.Get the constant term factors and the leading coefficient of the polynomial.. Find all possible rational roots.test each possible rational root until you find a root.. Factor the polynomial until you get a quadratic. (you can use synthetic division, quadratic formula, a calculator, or any other method).when using TI-8 or 85 store the polynomial in using the Y= menu. Store the root to be tested in. Use VARS Y-VARS 1: Function to evaluate What are therationalroots of 7 6 0 Y 1 Y 1 Did you remember what is a conjugate number? a bi and a bi are conjugates a b and a b are conjugates If a comple number or an irrational number is a root of a polynomial equation with rational coefficients, so is its conjugate. Conjugate Root Theorem: If () is a polynomial with rational coefficients, then the irrational roots of ()=0 that have a form a b occur in conjugate pairs. That is, If a b is an irrational root with rational then a b is also a root. If () is a polynomial with real coefficients, then the comple roots of ()=0 occur in conjugate pairs. That is a bi is a comple root with a and b real, then a bi is also a root.
roblem :Using the Conjugate Root Theorem to Identify Roots. A quartic polynomial Answer: has rational coefficients. If Since () has rational coefficients and 0 and1 i are roots of () 0 what are theother two roots? isa root of () 0, it follows from the Conjugate Root Theorem that 0 - isalsoa root. Since() has real coefficients and1 i isaroot od () 0, it follows that 1-i isalsoa root. Theother two roots are - and 1 i A cubic polynomial () has real coefficients. If -i 5 and are two roots of () 0, What istheadditional root? roblem : Using the Conjugates to construct A olynomial A. What is a third roots - and i? deg ree polynomial function y () with rational coefficients so that () 0 has the B. What quartic polynomial equation has roots -i, 8,? Descartes Rule of Signs Theorem: Let () be a polynomial with real coefficients written in standard form. *The number of positive real roots of () is either equal to the number of sign changes between consecutive coefficients of () or is less than that by an even number. *The number of negative real roots of ()=0 is either equal to the number of sign changes between consecutive coefficients of (-) or is less than that by an even number. In both cases, count multiple roots according to their multiplicity This theorem implies that two tests must be done on a polynomial function to determine both positive and negative real roots.
roblem 5: Using the Descartes Rule of Signs. What the Descartes' Rule of Signs tell you about the real roots of 1 0 There are two sign changes, + to and - to +. Therefore, there are either 0 or positive real roots. To find the negative real root you need to plug in into the polynomial There is only one negative real root 1 0 Take a note: Graph the function and recall that cubic functions have zero or two turning points. Because the graph already shows two turning points, it will not change the direction again. So there are no positive real roots. A.What does Descarte s' Rule of Signs tell you about the real roots of 1 0 1 0 B.Can you confirm re al and comple roots graphically? Eplain Take a note: the degree of a polynomial equation tells you how many roots the equation has. That is the result of the Fundamental theorem of Algebra provided by the German mathematician Carl Friedrich Gauss (1777-1855) The fundamental Theorem of Algebra: If () is a polynomial of degree n 1, then ()=0 has eactly n roots, including multiple and comple roots. roblem 6: Using the Fundamental Theorem of Algebra What are all the roots of 5 0 I need to find the zeros(5) and using the rational root and factor theorem, synthetic division and factoring Using the rational root the possible rational roots are 1,, Evaluate the polynomial function for 1 (1) 0,1 isa root and ( 1) isa factor Then use synthetic division 5 1 1-1 - - 1 1 0-0 - The factors are - and 1 1 0-0 - 0 then 1 1 1 0 1 1 i i Answer: (-1)(-)(+)(-i)(+i) so the roots are 1,-,,i,-i
What are all theroots of the equation 1 10? Since there is no constant term, make the equation equal to zero and factor from the polynomial roblem 7: Finding all the zeros of a olynomial Function What are the zerosof f() 7 9 18 Step 1: use the graphing calculator to find any real roots Step : Factor out the linear factors (-)(+).Use Synthetic division twice. Step: use the quadratic formula to find the comple roots Step : What are all the zerosof the function g( ) 6? The Fundamental Theorem of Algebra: Here are equivalent ways to state the fundamental theorem of Algebra. You can use any of these statements to prove the others. *Every polynomial equation of degree n 1 has eactly n roots, including multiple and comple roots. *Every polynomial of degree n 1 has n factors. *Every polynomial function of degree n 1 has at least one comple zero.