Lesson 6 Zero Product Law - Weebly

The Zero Product Law Objective:– Students will be able to (SWBAT) use complex numbers inpolynomial identities and equations, in order to (IOT) solvequadratic equations with real coefficient that have complexsolutions. Standards:– MGSE9-12.N.CN.7 – Solve Quadratics. – Solve quadratic equations with realcoefficients that have complex solutions by (but not limited to) square roots,completing the square, and the quadratic formula.

The Zero Product LawBell Ringer: What are the solutions to the equation 2" # " 1 0?)A. *)B. *)C. *)D. * *) and *,*),and **,**)- and * *)- and *,* *-The correct answer choice is (c).-

Factoring Trinomials Factoring trinomials, expressions of the form !" # %" &, is animportant skill. Trinomials can be factored if they are the product oftwo binomials. The two main keys to factoring trinomials are: (1) the ability to quicklyand accurately multiply binomials (FOIL) and (2) the ability to workwith signed numbers.

Factoring Trinomials We practice both of these skills with four warm-up multiplicationproblems in Exercise #1. Exercise #1: Without using your calculator, write each of the followingproducts in simplest !" # %" & form.(a) 3" 2 5" 7(b) 2" 3 2" 5(c) 5" 4 " 2(d) 4" 3 3" 8

Factoring Trinomials It is important that you know the fundamental rules governing themultiplication and addition of signed numbers. These rules will be key in factoring quickly and correctly. In each case, where a trinomial can be factored, it will be done usingthe guess-and-check method, where we intelligently guess binomialpairs and then check by seeing if the linear terms of the multiplicationcombine to form the linear term of the trinomial.

Factoring Trinomials Exercise #2: Consider the trinomial 6" # 35" 6. Below are four guesses ofhow this trinomial factors.3" 2 2" 3" 3 " 26" 1 " 63" 2 2" 3(a) Two of these guesses are “unintelligent” – meaning that they should not even bechecked. Cross them out and explain below them why they are unreasonable.(b) Of the two that remain, check both above and determine which is the correctfactorization of the trinomial.

Factoring Trinomials The easiest of all trinomial factoring occurs when the leadingcoefficient is one ! 1 . Exercise #3: Using a guess-and-check technique, factor each of thefollowing trinomials.(a) % 2 35(b) % 11 24(c) % 13 22(d) % 5 50

Factoring Trinomials A step up from the last exercise occurs when the leading coefficientisn’t one but is still a prime number. This is very often the case and makes at least part of the guessingmuch easier. Exercise #4: Using a guess-and-check technique, factor each of thefollowing trinomials that have prime leading coefficients. Show eachguess and its check.(a) 3" # 19" 40(b) 2" # 15" 18

Factoring Trinomials Finally, the hardest trinomials to factor are those whose leadingcoefficients are not prime. This is due to the fact that there are so many more intelligent guesses. In future lessons we will develop ways to eliminate some of these, butfor now, the key will be to just keep guessing until you get it right.

Factoring Trinomials Exercise #5: Factor each of the following trinomials. Show eachguess and its check.(a) 15# 13# 2(b) 10# 13# 30

The Zero Product Law One of the most important equation solving technique stems from afact about the number zero that is not true of any other number:THE ZERO PRODUCT LAWIf the product of multiple factors is equal to zero then at least one of thefactors must be equal to zero.

The Zero Product Law The law can immediately be put to use in the first exercise. In this exercise,quadratic equations are given already in factored form. Exercise #1: Solve each of the following equations for all value(s) of x.(a) ! 7 ! 3 0(b) 2! 5 ! 4 0 (c) 4 3! 2 4! 3 0

The Zero Product Law Exercise #2: In Exercise #1(c), why does the factor of 4 have no effecton the solution set of the equation?The Zero Product Law states that if a factored equation is equal to zero then atleast one of the factors must be zero. It does not mean all factors must be zero.Hence, if a factor is a constant, then it has no effect on the solution set. The Zero Product Law can be used to solve any quadraticequation that is factorable (not prime). To utilize thistechnique the problem solver must first set the equationequal to zero and then factor the non-zero side.

The Zero Product Law Exercise #3: Solve each of the following quadratic equations using theZero Product Law.(a) ! " 3! 14 2! 10(b) 3! " 12! 7 ! " 3! 2

The Zero Product Law Exercise #4: Consider the system ofequations shown below consisting of aparabola and a line.y! 3 % 8 5 and ! 4 5(a) Find the intersection points of thesecurves algebraically.x(b) Using your calculator, sketch a graph of this system on the axes to theright. Be sure to label the curves with equations, the intersection points,and the window.(c) Verify your answers to part (a) by using the INTERSECT command onyour calculator.

The Zero Product Law The Zero Product Law is extremely important infinding the zero’s or !-intercepts (zeroes) of aparabola. Exercise #5: The parabola shown at the righthas the equation " % 2 3.(a) Write the coordinates of the two -intercepts of the graph.(b) Find the -intercepts of this parabola algebraically.

The Zero Product Law Exercise #6: Algebraically find the set of x-intercepts (zeroes) for eachparabola given below.(a) ! 4 % 1(b) ! 3 % 13 10(c) ! 5 % 10

The Zero Product LawAcademic Agenda: Complete the The Zero Product Law Assignment. Closure: Exit Ticket – ReasoningA quadratic function of the form ! # &# '.(a) What are the x-intercepts of this parabola?(b) Based on your answer to part (a), write the equation of this quadraticfunction first in factored form and then in trinomial form. Homework: Review class notes and finish the The Zero Product Law Assignment.

The Zero Product Law Closure: Exit Ticket – ReasoningA quadratic function of the form ! # &# '.(a) What are the x-intercepts of this parabola?(b) Based on your answer to part (a), write theequation of this quadratic function first in factoredform and then in trinomial form.

Factoring Trinomials Factoring trinomials, expressions of the form !"# %" &, is an important skill. Trinomials can be factored if they are the product of two binomials. The two main keys to factoring trinomials are: (1) the ability to quickly and accurately multiply binomials (FOIL) and (2) the ability to work with signed numbers.