Multiplicities of polynomials

Multiplicities of polynomials

Lessons

The multiplicity of a zero corresponds to the number of times a factor is repeated in the function.
\cdot Odd multiplicity: cross the x-axis
\cdot Odd multiplicity (3 or more): changes concavity when passing through x-axis
\cdot Even multiplicity: bounces off the x-axis
  • 1.
    Sketch the graph of the polynomial function.
    i)
    P(x)=x2P\left( x \right) = x - 2
    ii)
    P(x)=(x2)2P\left( x \right) = {\left( {x - 2} \right)^2}
    iii)
    P(x)=(x2)3P\left( x \right) = {\left( {x - 2} \right)^3}
    iv)
    P(x)=(x2)4P\left( x \right) = {\left( {x - 2} \right)^4}
    v)
    P(x)=(x2)5P\left( x \right) = {\left( {x - 2} \right)^5}

  • 2.
    Given that the graph shows a degree-eight polynomial and the zero x=3x = 3 has a multiplicity of 2, determine the multiplicity of the zero x=2x = - 2.
    Multiplicities of polynomials

  • 3.
    Without using a graphing calculator, make a rough sketch of the following polynomial: P(x)=112907776(x+3)(x+1)2(x2)3(x4)4(x7)5P\left( x \right) = - \frac{1}{{12907776}}\left( {x + 3} \right){\left( {x + 1} \right)^2}{\left( {x - 2} \right)^3}{\left( {x - 4} \right)^4}{\left( {x - 7} \right)^5}