Polynomials_Zeros_Packet
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Name Date Block Honors Algebra II: Introduction to Polynomial Functions and Polynomial Zeros
Name Date Period Honors Algebra II: Introduction to Polynomial Functions Definition: A polynomial function is a function of the form: f(x)=a x"+a, x"* +...+ax +ax+a, , Where a,, a,..,a, are real numbers and n, n=1 is a natural number. The domain of a polynomial function is (-oo, oo). Determine if each of the following functions are polynomial functions. a) g(x)=2 b) h(x)= );2;24 ) Jj(x)=vx d) k(x)=15x*-3x*-10 Definitions: Given f(x)=ax"+a,,x"" +..+a3,x*+ax+a, with a =0, we say: - The natural number nis called the degree of the polynomial f. - Theterm ax" is called the leading term of the polynomial f. - The real number a, is called the leading coefficient of the polynomial f. - The real number g, is called the constant term of the polynomial f. Do together: Determine the degree, leading term, leading coefficient and constant term of each of the polynomials below. f(x)=10—2x3+5x4+x2—8x g(x):(Zx—l)(x+3)(5—x) degree: leading term: leading coefficient: constant term. Determine the degree, leading term, leading coefficient and constant term of each of the polynomials below. P(x)=-8X°—4x* 42 +x-9 | t(x)=12-8x"+2x* | p(x)=(4-x)(x +2)2 v(x)=-2x(x - 4)(2x +1)(x-3] degree: leading term. leading coefficient: constant term:
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Polynomial Long Division Examples of Long Division of Polynomial Functions . 3x2—15 x+ 5)3x° + 5x2 — 15x — 25 ~(3x? + 5x2) 0 — 15x — 25 ~(=15x — 25) 0 x + 5)3x3 + 22x% + 38x + 15 3x - 1)3x3 —5x2_26x+8 x—2)3x3+x2—12x—4 Is x-2 afactorof 3x* +x2-12x-47? Given p(x) =3x%+ x> -12x - 4, what is the value of p(z) ?
Homework: below. 1) Determine the degree, leading term, leading coefficient and constant term of each of the polynomials Fx)=-3x"+2x"+8 | g(x)=5(2x+1) (3-x) h(x) = (x—2)3(x+1)(x+5) k(x)= 2x(x—7)(1o—x) degree: leading term. leading coefficient: constant term. 2) Determine if each of the following are polynomial functions. If your answer is ‘no’ justify your decision. 2) m(X)=2x3+x‘}—5x—3 b) n(x)=2x3+);)—(5x—3 c) q(x)=2)‘2‘3”“28 d) r(x)=5x—2 &) t(x)=(x+1) ) r(e)=t'-16 3) Answer each of the following. Show the work/expla in the reasoning that leads to each of your answers. a)Given (x -3)(x +5)(2x +1) =2x* +5x* - 28x 15, 2x3 +5x%-28x-15 (Zx . 1) equal? what does b) Is x =4 a zero of the polynomial F(x)=x*-2x> -13x* +14x +247?
¢) What are the zeros of the polynomial function d) Given the function g(x) =2x>-5x*—x+6,is 2 P(X) = (3X +1) (X - 2)(X +1) ? x = -5 a zero of the function? Justify your answer. 4) Perform each of the following long division problems. Then use your results to answer the question that follows. a) b) 2x+1) 2x3—x*-61x-30 x—7) 23+ x*-13x+6 Is 2x +1 a factor of 2x3®-x?2-61x-307? Is x-7 afactor of 2x3+x*-13x+67?
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Date Period Name Honors Algebra II Intro to Polynomials: Synthetic Division & Rational Root Test Synthetic Division is an abbreviated version of long division that can be used when dividing by linear factors. 6x3+11x%2-17x—-30 x+2 Consider the following: We can determine the result with ... PRy . 2 H nsAct . Long division: G -x-15 Synthetic Division: 12 2 30 x+2)6x°+11x* - 17x - 30 (6x3+12x2) 6 -1 -5 0 - x*-17x Lol -15x-30 (~15x - 30) 0 Then once we get to a reminder of zero and a quotient that is quadratic, we have the tools to find all the zeros and write as a product of linear factors! f(x)-=6x3+1‘ix2-17x-30= with zeros of x =
Another example of synthetic division: Determine if 2x -1 is a factor of p(x) = 2x% +17x% + 31x - 20. If 2x -1 is a factor, then x =1§ is a zero, sO... 2 17 3 -2 N = Thus we can write the function in factored form as p(x) = ,» and the zeros are X = Now try using synthetic division to answer these questions: Is x+5 a factor of x3-4x2_7x+107? Is x.+7 afactor of 2x®.9x2-38x-21?
Rational Root Theorem is a way to determine the possible rational roots of a polynomial function. This theorem states that if there are rational roots, they are of the form » P, where pis a factor of the constant term q and gis a factor of the leading coefficient. Practice: For each of the following, list all the possible rational roots of each function. How many possible rational roots does each function have? a) f(x)=x°-4x>+2x* - x+12 b) g(x)=2x®+4x*-5x+10 ) h(x)=4x’-8x*+x-6 Finding Zeros of a Polynomial Function Now that we know how to find the possible zeros using the rational root test, let’s put it all together. Consider the polynomial function p(x) = 2x% 4 x3 £12%% - 34x - 20. Possible rational roots: Use synthetic division to find a zero/root: Now we can use what we know about quadratics to find the remaining zeros of the original cubic function. The zeros of the function are: Function in factored form
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Name Date Period Honors Algebra II Intro to Polynomials: Synthetic Division & Rational Root Test (WS2) 1) Determine if x + 3 is a factor of each of the polynomials given below. If it is a factor, write the polynomial as a product of linear factors and list the zeros. If it is not a factor, explain/show how your know this. a) g(x) =6x3+29x%-7x-10 b) h(x)=x3+2x2-11x-12 2) For each of the following, list the possible rational roots. Then use synthetic division to find the rational roots. Finally list all the zeros of the function and write the function in factored form (you will also need to use what you know about solving quadratics). Show your work in a neat and organized manner. a) Polynomial p(x) =x*+6x2+13x+20 Possible Rational Roots: Factored Form with linear factors: Factored form with linear factors:
b) Polynomial r(x) =x%-19x +30 hint: there are zero x squared terms! Possible Rational Roots: Factored Form with linear factors: Factored form with linear factors: ¢) Polynomial t(x)= x3+ x2-22x-40 Possible Rational Roots: Factored Form with linear factors: Factored form with linear factors:
d) Polynomial s(x)= 2x* - 7x3 - 3x% + 5x -1 Possible Rational Roots: Factored Form with linear factors: Factored form with linear factors: e) Polynomial v(x) =xt+ x5-31x*-61x-30 Possible Rational Roots: Factored Form with linear factors: Factored form with linear factors:
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Name Date Period Honors Algebra II Finding Zeros of Polynomials WS3 Write each of the polynomials below in factored form and list all of the zeros. 1) f(x)= 2x% +7x* —68x+32 Factored form: Zeros: 2) f(x)=x"- 5x°-2x*-20x-24 Factored form: Zeros:
3) fF(xX)=x3+x*+9x+9 Factored form: Zeros: 4) f(x)=x"-5x>-2x*+14x-20 Factored form: Zeros: 5) f(x)= x3-8x>+x+42 Factored form: Zeros: