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Polynomials

Math 122. Polynomials. Polynomials. A polynomial is any function that has an equation of the form y=c 0 +c 1 x+c 2 x 2 +…+c n x n where the powers of x must be integers and the letters c 0 …c n are numbers. The degree of a polynomial is the largest power of x in the polynomial.

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Polynomials

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  1. Math 122 Polynomials

  2. Polynomials • A polynomial is any function that has an equation of the form y=c0+c1x+c2x2+…+cnxn where the powers of x must be integers and the letters c0…cn are numbers. • The degree of a polynomial is the largest power of x in the polynomial. • The factored form of a polynomial is :y=k(x-r1)m1(x-r2)m2…(x-rn)mn where k is

  3. Some Examples • y=3x2+2 • Is this a polynomial? • Yes • What is its degree? • 2 • y=5x+x(1/2) • Is this a polynomial? • No (Why?)

  4. Some More Examples • y=-3 • Is this a polynomial? • Yes • What is its degree? • 0 • y=(x+2)(x-1)2 • Is this a polynomial? • Yes • What is its degree? • 3 (Notice the sum of the exponents of the factors)

  5. Common Polynomials

  6. A bit about polynomials • The factored form of a polynomial is: y=k(x-r1)m1(x-r2)m2…(x-rn)mn where • k is called the constant of proportionality • r1…rn are the roots of the polynomial • The roots of a polynomial are the x-intercepts • m1…mn are the multiplicities of the roots • The multiplicities of a polynomial are determined by how the graph hits the x-axis

  7. Determining Multiplicity One • A root with muliplicity one occurs when the graph goes straight through the x-axis. • In this graph, the roots x=2 and x=-2 have multiplicity one.

  8. Determining Multiplicity Two • A root with multiplicity two occurs when the graph touches the x-axis and “turns around”. Like the vertex of a parabola. • In this graph the root x=1 has multiplicity 2.

  9. Determining Multiplicity Three • A root with multiplicity three has an inflection point on the x-axis. (It has a slight curve) • In this graph the root x=-1 has a multiplicity of three.

  10. Finding the Equation of a Polynomial • Locate the x-intercepts (roots). • Determine the multiplicity of each root. • Write the polynomial in factored form. • Solve for k, the constant of proportionality, by using another point on the graph. (don’t use a root)

  11. Example • Roots: -3, -1, 2 • Mult: 1, 2, 3 • Factored form: y=k(x+3)1(x+1)2(x-2)3 • Use the point (0,-2): • -2=k(3)(1)2(-2)3 • -2=-24k, so k=1/12 • y=(1/12)(x+3)(x+1)2(x-2)3

  12. The End By Marcy Gertsen

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