Find All The Zeros Of $f(x) = X^3 - 17x^2 + 120x - 350$ Given That $5 - 5i$ Is A Zero.If The Answer Contains An Imaginary Part, Enter The Answer In The Standard Form Of A Complex Number, \$a + Bi$[/tex\]. If

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Introduction

In algebra, finding the zeros of a polynomial is a crucial step in understanding its behavior and properties. Given a cubic polynomial, we can use various methods to find its zeros, including factoring, synthetic division, and the Rational Root Theorem. However, when one of the zeros is a complex number, we need to employ more advanced techniques to find the other zeros. In this article, we will explore how to find the zeros of a cubic polynomial when one of the zeros is a complex number.

The Problem

We are given a cubic polynomial:

f(x)=x3βˆ’17x2+120xβˆ’350f(x) = x^3 - 17x^2 + 120x - 350

and we know that one of its zeros is a complex number:

5βˆ’5i5 - 5i

Our goal is to find all the zeros of the polynomial.

Complex Conjugate Root Theorem

The Complex Conjugate Root Theorem states that if a polynomial with real coefficients has a complex zero, then its complex conjugate is also a zero of the polynomial. In this case, the complex conjugate of 5βˆ’5i5 - 5i is 5+5i5 + 5i. Therefore, we can conclude that 5+5i5 + 5i is also a zero of the polynomial.

Factor Theorem

The Factor Theorem states that if aa is a zero of a polynomial f(x)f(x), then (xβˆ’a)(x - a) is a factor of f(x)f(x). In this case, we know that 5βˆ’5i5 - 5i is a zero of the polynomial, so (xβˆ’(5βˆ’5i))(x - (5 - 5i)) is a factor of f(x)f(x). Similarly, we know that 5+5i5 + 5i is a zero of the polynomial, so (xβˆ’(5+5i))(x - (5 + 5i)) is also a factor of f(x)f(x).

Factoring the Polynomial

We can use the Factor Theorem to factor the polynomial:

f(x)=(xβˆ’(5βˆ’5i))(xβˆ’(5+5i))(xβˆ’r)f(x) = (x - (5 - 5i))(x - (5 + 5i))(x - r)

where rr is the third zero of the polynomial.

Expanding the Factors

We can expand the factors to get:

f(x)=(x2βˆ’10x+50)(xβˆ’r)f(x) = (x^2 - 10x + 50)(x - r)

Expanding the Product

We can expand the product to get:

f(x)=x3βˆ’rx2βˆ’10x2+10rx+50xβˆ’50rf(x) = x^3 - rx^2 - 10x^2 + 10rx + 50x - 50r

Combining Like Terms

We can combine like terms to get:

f(x)=x3βˆ’(r+10)x2+(10r+50)xβˆ’50rf(x) = x^3 - (r + 10)x^2 + (10r + 50)x - 50r

Equating Coefficients

We can equate the coefficients of the polynomial with the given polynomial:

x3βˆ’17x2+120xβˆ’350=x3βˆ’(r+10)x2+(10r+50)xβˆ’50rx^3 - 17x^2 + 120x - 350 = x^3 - (r + 10)x^2 + (10r + 50)x - 50r

Solving for r

We can solve for rr by equating the coefficients of the x2x^2 terms:

βˆ’17=βˆ’(r+10)-17 = -(r + 10)

Solving for rr, we get:

r=7r = 7

Finding the Third Zero

We can substitute r=7r = 7 into the factored form of the polynomial:

f(x)=(xβˆ’(5βˆ’5i))(xβˆ’(5+5i))(xβˆ’7)f(x) = (x - (5 - 5i))(x - (5 + 5i))(x - 7)

Expanding the Product

We can expand the product to get:

f(x)=(x2βˆ’10x+50)(xβˆ’7)f(x) = (x^2 - 10x + 50)(x - 7)

Expanding the Product

We can expand the product to get:

f(x)=x3βˆ’7x2βˆ’10x2+70x+50xβˆ’350f(x) = x^3 - 7x^2 - 10x^2 + 70x + 50x - 350

Combining Like Terms

We can combine like terms to get:

f(x)=x3βˆ’17x2+120xβˆ’350f(x) = x^3 - 17x^2 + 120x - 350

Conclusion

We have found all the zeros of the cubic polynomial:

f(x)=x3βˆ’17x2+120xβˆ’350f(x) = x^3 - 17x^2 + 120x - 350

The zeros are:

5βˆ’5i5 - 5i

5+5i5 + 5i

77

These zeros can be used to factor the polynomial and understand its behavior.

Final Answer

The final answer is:

\boxed{7}$<br/> # Q&A: Finding the Zeros of a Cubic Polynomial with a Given Complex Zero =========================================================== ## Introduction --------------- In our previous article, we explored how to find the zeros of a cubic polynomial when one of the zeros is a complex number. We used the Complex Conjugate Root Theorem and the Factor Theorem to find the other zeros of the polynomial. In this article, we will answer some frequently asked questions related to finding the zeros of a cubic polynomial with a given complex zero. ## Q: What is the Complex Conjugate Root Theorem? -------------------------------------------- A: The Complex Conjugate Root Theorem states that if a polynomial with real coefficients has a complex zero, then its complex conjugate is also a zero of the polynomial. This means that if a polynomial has a complex zero, its complex conjugate will also be a zero of the polynomial. ## Q: How do I find the complex conjugate of a complex number? -------------------------------------------------------- A: To find the complex conjugate of a complex number, you need to change the sign of the imaginary part. For example, the complex conjugate of $5 - 5i$ is $5 + 5i$. ## Q: What is the Factor Theorem? ----------------------------- A: The Factor Theorem states that if $a$ is a zero of a polynomial $f(x)$, then $(x - a)$ is a factor of $f(x)$. This means that if a polynomial has a zero, you can factor the polynomial using that zero. ## Q: How do I use the Factor Theorem to factor a polynomial? --------------------------------------------------------- A: To use the Factor Theorem to factor a polynomial, you need to find the zeros of the polynomial and then use those zeros to factor the polynomial. For example, if you know that $5 - 5i$ is a zero of the polynomial, you can use the Factor Theorem to factor the polynomial as $(x - (5 - 5i))(x - r)$, where $r$ is the other zero of the polynomial. ## Q: What is the difference between a complex zero and a real zero? --------------------------------------------------------- A: A complex zero is a zero that has an imaginary part, while a real zero is a zero that has no imaginary part. For example, $5 - 5i$ is a complex zero, while $7$ is a real zero. ## Q: Can a polynomial have only real zeros? ----------------------------------------- A: Yes, a polynomial can have only real zeros. For example, the polynomial $x^3 - 7x^2 + 12x - 6$ has only real zeros. ## Q: Can a polynomial have only complex zeros? ----------------------------------------- A: No, a polynomial cannot have only complex zeros. This is because complex zeros always come in conjugate pairs, so there must be at least one real zero. ## Q: How do I find the zeros of a polynomial with complex coefficients? ---------------------------------------------------------------- A: To find the zeros of a polynomial with complex coefficients, you need to use the quadratic formula or other methods to find the zeros of the polynomial. However, if the polynomial has real coefficients, you can use the Complex Conjugate Root Theorem to find the zeros of the polynomial. ## Q: What is the significance of finding the zeros of a polynomial? --------------------------------------------------------- A: Finding the zeros of a polynomial is important because it allows you to understand the behavior of the polynomial and its roots. The zeros of a polynomial can be used to factor the polynomial, find the maximum and minimum values of the polynomial, and solve equations involving the polynomial. ## Conclusion -------------- In this article, we have answered some frequently asked questions related to finding the zeros of a cubic polynomial with a given complex zero. We have discussed the Complex Conjugate Root Theorem, the Factor Theorem, and the significance of finding the zeros of a polynomial. We hope that this article has been helpful in understanding the concepts of complex zeros and polynomial factorization. ## Final Answer -------------- The final answer is: * The Complex Conjugate Root Theorem states that if a polynomial with real coefficients has a complex zero, then its complex conjugate is also a zero of the polynomial. * The Factor Theorem states that if $a$ is a zero of a polynomial $f(x)$, then $(x - a)$ is a factor of $f(x)$. * A complex zero is a zero that has an imaginary part, while a real zero is a zero that has no imaginary part. * A polynomial can have only real zeros or complex zeros, but not both. * Finding the zeros of a polynomial is important because it allows you to understand the behavior of the polynomial and its roots.