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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Aim: How do we find the zeros of polynomial functions? Do Now: A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. General Features of a Polynomial Function Describe some basic characteristics of this polynomial function: Continuous no breaks in curve Smooth no sharp turns discontinuous sharp turn NOT POLYNOMIAL FUNCTIONS DEFINITION: Polynomial Function Let n be a nonnegative integer and let a 0, a 1, a 2,... a n-1, a n be real numbers with a n 0. The function given by f(x) = a n x n + a n - 1 x n a 2 x 2 + a 1 x + a 0 is a polynomial function of degree n. The leading coefficient is a n. The zero function f(x) = 0 is a polynomial function. It has no degree and no leading coefficient.

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. General Features of a Polynomial Function Leading Coefficient Cubic term Quadratic term Linear term Constant term Standard Form Degree Polynomial of 4 terms

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. General Features of a Polynomial Function Simplest form of any polynomial: y = x n n > 0 When n is even looks similar to x 2 When n is odd looks similar to x 3 The greater the value of n, the flatter the graph is on the interval [ -1, 1].

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Transformations of Higher Degree Polys If k and h are positive numbers and f(x) is a function, then f(x ± h) ± k shifts f(x) right or left h units shifts f(x) up or down k units f(x) = (x – h) 3 + k - cubic f(x) = (x – h) 4 + k - quartic ex. f(x) = (x – 4) 4 – 2

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Zeros of Polynomial Functions The zero of a function is a number x for which f(x) = 0. Graphically its the point where the graph crosses the x-axis. Ex. Find the zeros of f(x) = x 2 + 3x f(x) = 0 = x 2 + 3x = x(x + 3) x = 0 and x = -3 For polynomial function f of degree n, the function f has at most n real zeros How many roots does f(x) = x have? the graph of f has at most n – 1 relative extrema (relative max. or min.).

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number plane. Degree of polynomial FunctionZeros 1st n = 1 f(x) = x – 3x = 3 2nd n = 2 f(x) = x 2 – 6x + 9 = (x – 3)(x – 3) x = 3 and x = 3 3rd n = 3 f(x) = x 3 + 4x = x(x – 2i)(x + 2i) x = 0, x = 2i, x = -2i 4th n = 4 f(x) = x 4 – 1 = (x – 1)(x + 1)(x – i)(x + i) x = 1, x = -1, x = i, x = -i repeated zero

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Finding Zeros Find the zeros of f(x) = x 3 – x 2 – 2x f(x) = 0 = x 3 – x 2 – 2x = x(x 2 – x – 2) = x(x – 2)(x + 1) x = 0, x = 2 and x = -1 Has at most 3 real roots Has 2 relative extrema

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Finding a Function Given the Zeros Write a quadratic function whose zeros (roots) are -2 and 4. x = -2 x = 4 x + 2 = 0 x – 4 = 0 reverse the process used to solve the quadratic equation. (x + 2)(x – 4) = 0 x 2 – 2x – 8 = 0 x 2 – 2x – 8 = f(x) Find a polynomial function with the following zeros: -2, -1, 1, 2 f(x) = (x + 2)(x + 1)(x – 1)(x – 2) f(x) = (x 2 – 4)(x 2 – 1) = x 4 – 5x 2 + 4

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Multiplicity Find the zeros of f(x) = x 4 + 6x 3 + 8x 2. f(x) = x 4 + 6x 3 + 8x 2 A multiple zero has a multiplicity equal to the numbers of times the zero occurs.

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Regents Prep The graph of y = f(x) is shown at right. Which set lists all the real solutions of f(x) = 0? 1.{-3, 2} 2.{-2, 3} 3.{-3, 0, 2} 4.{-2, 0, 3}

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Model Problem Find the zeros of f(x) = 27x Factoring Difference/Sum of Perfect Cubes u 3 – v 3 = (u – v)(u 2 + uv + v 2 ) u 3 + v 3 = (u + v)(u 2 – uv + v 2 )

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Model Problem Find the zeros of f(x) = 27x

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Polynomial in Quadratic Form Find the zeros= 0 2 us – 4 zeros

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Finding zeros by Factoring by Groups f(x) = x 3 – 2x 2 – 3x + 6 Group terms (x 3 – 2x 2 ) – (3x – 6) = 0 Factor Groups x 2 (x – 2) – 3(x – 2) = 0 Distributive Property (x 2 – 3)(x – 2) = 0 Find the roots of the following polynomial function. x 3 – 2x 2 – 3x + 6 = 0 Solve for x x 2 – 3 = 0; x – 2 = 0; x = 2

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Aim: Roots of Polynomial Equations Course: Alg. 2 & Trig. Regents Prep Factored completely, the expression 12x x 3 – 12x 2 is equivalent to

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