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Chapter 2: Problem 47

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=2 x^{4}-19 x^{3}+51 x^{2}-31 x+5$$

Short Answer

Expert verified
The zeros of the polynomial are \(x = 1\), \(x = 1/2\), and \(x = 5\), with the zero \(x = 1/2\) having a multiplicity of 2 and the zeros 5 and 1 have a multiplicity of 1 each.

Step by step solution

01

Understanding the concept of zeros

The zero of a polynomial is the value of \(x\) that makes the polynomial equal to zero. To find the zeros, we set the \(P(x)\) equal to zero and solve for \(x\). The multiplicity of a zero is the number of times that root appears. It is also the exponent of that factor in the factored form of the polynomial.
02

Setting the Polynomial Equal to Zero

We begin by setting the polynomial equal to zero to find the roots. This gives us the equation: \(2x^{4} - 19x^{3} + 51x^{2} - 31x + 5 = 0\). The equation will produce 4 roots since it's a polynomial of order 4.
03

Factoring the polynomial

Factoring the polynomial can be quite complex as the order of polynomial is high. However, using synthetic division and through trial and error, we can find that the roots are \(x = 1, 1/2, 5, 1/2\). We factor the polynomial like this:\[2x^4 - 19x^3 + 51x^2 - 31x + 5 =2 (x-1) (2x-1) ^{2} (x-5)\]
04

Identifying the zeros and their multiplicities

From the factored form of the polynomial, we see that the zeros are \(x = 1, 1/2, 5\) and the zero \(x = 1/2\) has multiplicity 2, as it appears twice in the factored form, whereas the zeros 5 and 1 are simple zeros with multiplicity 1.

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Most popular questions from this chapter

The property that the product of conjugates of the form \((a+b i)(a-b i)\) is equal to \(a^{2}+b^{2}\) can be used to factor the sum of two perfect squares over the set of complex numbers. For example, \(x^{2}+y^{2}=(x+y i)(x-y i) .\) In Exercises 71 to \(74,\) factor the binomial over the set of complex numbers. $$x^{2}+9$$

In Exercises 61 to 70 , use the quadratic formula to solve each quadratic equation. $$4 x^{2}-4 x=-9$$

Find a polynomial function \(P(x)\) that has the indicated zeros. Zeros: \(2-5 i,-4 ;\) degree 3

In Exercises 61 to 70 , use the quadratic formula to solve each quadratic equation. $$8 x^{2}+12 x=-17$$

Find a polynomial function of lowest degree with integer coefficients that has the given zeros. $$-1,1,-5$$
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