A polynomial function #p# can be factored into seven factors: #( x-3), ( x+1)#, and 5 factors of #(x -2)#. What are its zeros with multiplicity, and what is the degree of the polynomial? Explain.

1 Answer
jessicatweed · Amory W.
Nov 29, 2017

Zero 3 has multiplicity 1, zero -1 has multiplicity 1, and zero 2 has multiplicity 5. The degree of the polynomial is 7.

Explanation:

#p(x)=(x-3)(x+1)(x-2)(x-2)(x-2)(x-2)(x-2)#

The multiplicity of an equation is how many times a zero repeats.

#p(x)=(x-3)(x+1)(x-2)^5#

In the equation, zero 3 has multiplicity 1, zero -1 has multiplicity 1, and zero 2 has multiplicity 5.

The degree of the whole polynomial is the highest degree out of every term.
So first you have to expand:
#p(x)=(x-3)(x+1)(x-2)(x-2)(x-2)(x-2)(x-2)#
#p(x)=(x^2+x-3x-3)(x^2-4x+4)(x^2-4x+4)(x-2)#
#p(x)=(x^2-2x-3)(x^2-4+4)(x^3-6x^2+12x-8)#
#p(x)=(x^4-6x^3+9x^2+4x-12)(x^3-6x^2+12x-8)#
#p(x)=x^7-12x^6+57x^5-130x^4+120x^3+48x^2-176x+96#

So the degree of #p(x)=(x-3)(x+1)(x-2)(x-2)(x-2)(x-2)(x-2)# is 7.

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