Question

find the zeros and give the multiplicity of each for ( f(x)=x^{3}(x - 4)^{4}(x + 7)^{2} ).
note: to be counted as correct, you must get all answers correct.

• one zero is ( x=square ) and has a multiplicity of ( square ). help (numbers)
• another zero is ( x=square ) and has a multiplicity of: ( square ). help (numbers)
• the last zero is ( x=square ) and has a multiplicity of: ( square ). help (numbers)

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Explanation:

Step1: Find the zeros

For a polynomial \(f(x)=x^{3}(x - 4)^{4}(x + 7)^{2}\), the zeros are found by setting each factor equal to zero.

• For \(x^{3}=0\), we get \(x = 0\).
• For \((x - 4)^{4}=0\), we get \(x=4\).
• For \((x + 7)^{2}=0\), we get \(x=-7\).

Step2: Determine the multiplicity

The multiplicity of a zero \(a\) for a polynomial \(P(x)=(x - a)^{n}Q(x)\) (where \(Q(a)
eq0\)) is \(n\).

• For the factor \(x^{3}\), the zero \(x = 0\) has multiplicity \(3\).
• For the factor \((x - 4)^{4}\), the zero \(x = 4\) has multiplicity \(4\).
• For the factor \((x + 7)^{2}\), the zero \(x=-7\) has multiplicity \(2\).

Answer:

The zero \(x = 0\) has a multiplicity of \(3\), the zero \(x = 4\) has a multiplicity of \(4\), and the zero \(x=-7\) has a multiplicity of \(2\).