Problem 46 Find the zeros of each polynomia... [FREE SOLUTION]

Chapter 2: Problem 46

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=x^{3}-7 x^{2}-7 x+69$$

Short Answer

Expert verified

The zeros of the polynomial $P(x)=x^{3}-7 x^{2}-7 x+69$ are $x=3$ with multiplicity 1 and $x=4$ with multiplicity 2.

Step by step solution

Write down the polynomial function

Start with the given function $P(x) = x^3 - 7x^2 - 7x + 69$. We are looking for values of $x$ (the zeros) that make this equation equal zero.

Switch polynomial to the zero form

A polynomial can be written in the form $a(x - h)^n$, where $h$ is a zero of the polynomial with multiplicity $n$. Equate the function to zero: $x^3 - 7x^2 - 7x + 69 = 0$. Now we need to solve for $x$ to find the zeros.

Factor the polynomial

Factor the polynomial to make it easier to solve. Here it can be factored as $(x-3)(x-4)^2=0$. Note: Factoring could be made easier by using suitable factoring techniques or software.

Solve for X

Set each factor equal to zero and solve for $x$. From $(x-3)(x-4)^2=0$, we get $x=3$ and $x=4$.

Find the multiplicity of each zero

The multiplicity of a zero $x = h$ is the exponent of $(x-h)$ in the factored form. Here, $x=3$ appears once, so its multiplicity is 1. $x=4$ has an exponent of 2 in the factored form, so its multiplicity is 2.

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91影视

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=x^{3}-7 x^{2}-7 x+69$$

Short Answer

Step by step solution

Write down the polynomial function

Switch polynomial to the zero form

Factor the polynomial

Solve for X

Find the multiplicity of each zero

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