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Chapter 2: Problem 58

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+5 x+1$$

Short Answer

Expert verified
The zero of the polynomial function \(P(x)=x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+5 x+1\) is \(x = -1\) and its multiplicity is 5.

Step by step solution

01

Recognize binomial structure

The polynomial can be rewritten as \((x+1)^5\), where every term corresponds to the binomial theorem for \(n=5\). That is \(P(x) = (x+1)^5\).
02

Solve for the zeros of the function

Set \((x+1)^5 = 0\) to find the zeros of the function. This equation will be true when \(x+1 = 0\). Solve for \(x\) to get \(x = -1\). The zero of the function is \(x = -1\).
03

Determine the multiplicity of the zero

Since the zero \(x = -1\) is raised to the power of 5 in the function \((x+1)^5 = 0\), the multiplicity of this zero is 5. The zero \(x = -1\) appears five times in the equation.

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Most popular questions from this chapter

Show that if \(x=1-2 i,\) then \(x^{2}-2 x+5=0\)

Find a polynomial function \(P(x)\) with real coefficients that has the indicated zeros and satisfies the given conditions. Zeros: -1,2,\(3 ;\) degree \(3 ; P(1)=12\)

Find a polynomial function \(P(x)\) that has the indicated zeros. Zeros: \(3+2 i, 7 ;\) degree 3

In Exercises 61 to 70 , use the quadratic formula to solve each quadratic equation. $$4 x^{2}+4 x+5=0$$

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