P.o.D. – Factor each of the following: 1.) 2 + 10 + 21 2.) 2 − 3 − 4 3.) 2 − 15 + 54 4.) 2 + 12 + 32 5.) 2 − 100 1.) 2.) 3.) 4.) (x+3)(x+7) (t-4)(t+1) (x-6)(x-9) (p+8)(p+4) 5.) (x+10)(x-10) 11-4: The Factor Theorem Learning Target(s): I can find zeros of polynomial functions by factoring; apply the zeroproduct theorem and the factor theorem; graph polynomial functions; estimate zeros using graphs. Zero(s) of a Polynomial: - Places where the graph crosses the x-axis - Solutions to the equation - Also known as a root Zero Product Theorem: For all a and b, ab=0 if and only if a=0 or b=0. EX: An open box has sides of length x, (24-2x), and (24-2x). Thus, its volume is given by V(x)=x(24-2x)(24-2x). Find the zeros of V. The Zero-Product Theorem says that in order for an equation to equal zero, each of its factors could equal zero. X=0 24-2x=0 24-2x=0 24=2x 24=2x 12=x 12=x Solutions: 0,12,12 12 is a double root Factor Theorem: (x-r) is a factor of P(x) if and only if P(r)=0. In other words, r is a zero of P. EX: Find the roots of ( ) = 4 − 3 − 20 2 by factoring. Begin by taking out a common factor. ( ) = 2 ( 2 − − 20) Now factor the trinomial by un-FOILing. ( ) = 2 ( − 5)( + 4) Use the Zero-Product Property to set each factor equal to zero and solve. X+4=0 2 = 0 x-5=0 X=-4 = +0, −0 x=5 The zeros are x=0,0,5,-4. O is a double root. EX: Find the roots of ( ) = 4 − 14 2 + 45 Since this has 3 terms, we should try to un-FOIL. ( ) = ( 2 − 9)( 2 − 5) Notice that one of these factors is the difference of squares. ( ) = ( − 3)( + 3)( 2 − 5) Apply the zero-product property. x-3=0 x=3 x+3=0 x=-3 2 − 5 = 0 2 = 5 = ±√5 The roots are 3, -3, √5, −√5. This is an example of the Fundamental Theorem of Algebra. The FTA states that you will have the same number of roots as the highest exponent. EX: A polynomial function p with degree 4 and leading coefficient 1 is graphed below. Find the factors of p(x) and use them to write a formula for p(x). This graph crosses the x-axis, or has zeros, at -7, -4, 0, and 3. Therefore, it has factors of (x+7), (x+4), x, and (x-3). Expand these factors to find the polynomial. ( ) = ( + 7)( + 4)( − 3) = ( 2 + 7)( + 4)( − 3) = ( 3 + 4 2 + 7 2 + 28)( − 3) = ( 3 + 11 2 + 28)( − 3) = 4 + 11 3 + 28 2 − 3 3 − 33 2 − 84 = 4 + 8 3 − 5 2 − 84 EX: Find the equation of a polynomial whose zeros are -4, 7/2, and 5/3. The factors are (x+4), (x-7/2), and (x-5/3). 7 5 ( ) = ( + 4) ( − ) ( − ) 2 3 Or ( ) = ( + 4)(2 − 7)(3 − 5) 7 5 = ( − + 4 − 14) ( − ) 2 3 1 5 2 = ( + − 14) ( − ) 2 3 2 1 2 5 2 5 70 = + − 14 − − + 2 3 6 3 3 7 2 89 70 = − − + 6 6 3 Or ( ) = 6 3 − 7 2 − 89 + 140 3 Upon completion of this lesson, you should be able to: 1. Find zeros of a polynomial. 2. Find the roots of a polynomial by factoring. 3. Use ZoomMath (not on ACT). For more information, visit http://www.purplemath.com/modules/from zero2.htm HW Pg.756 2-12, 14-27 Quiz 11.1-11.4 tomorrow

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