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Chapter 2: Problem 59

Finding the Zeros of a Polynomial Function, use the given zero to find all the zeros of the function. $$Function$$ $$g(x)=4 x^{3}+23 x^{2}+34 x-10$$ $$Z \mathrm{er} 0$$ $$-3+i$$

Short Answer

Expert verified
The zeros of the given polynomial function \(g(x)=4x^{3}+23x^{2}+34x-10\) are \(-3 + i\), \(-3 - i\) and \(1/4\).

Step by step solution

01

Understand the Problem and Given Information

We have a polynomial function \(g(x)\) given by \(g(x)=4x^{3}+23x^{2}+34x-10\), and it is known that one of its zeros is \(x=-3+i\). This is a complex number, so its conjugate \(x=-3-i\) must also be a zero of the function since the coefficients of the polynomial are real.
02

Utilize Given Zeros to Factor the Polynomial

Given zeros are \(x = -3 + i\) and its conjugate \(x = -3 - i\), hence, the corresponding factors will be \(x - (-3 + i) = x + 3 - i\) and \(x - (-3 -i) = x + 3 + i\). Multiply these two factors which results in a quadratic polynomial \(x^{2} + 6x + 10\) which is a factor of the original cubic polynomial.
03

Find the Remaining Factor

Apply polynomial division to the original cubic polynomial \(g(x)\) by the quadratic polynomial \(x^{2} + 6x + 10\) we obtained earlier. The quotient we get will be the third factor, \(4x - 1\).
04

Find the Third Zero

Set \(4x - 1 = 0\), the third zero is \(x = 1/4\). Now we have all the zeros of the polynomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Conjugates
When dealing with polynomials that have real coefficients, the presence of a complex zero necessarily means its complex conjugate is also a zero.
For example, if you are given a zero like \(-3 + i\), you can immediately know that the complex conjugate, \(-3 - i\), is also a zero for the polynomial.
This concept is rooted in the nature of polynomial equations with real coefficients.
  • If a polynomial equation has real coefficients, complex roots will always appear in conjugate pairs.
  • This is crucial since it ensures that when multiplied out, the imaginary parts cancel, leaving only real numbers.
When using complex conjugates, realize that the polynomial can be expressed in terms of these factors that are actually quadratic in terms of real numbers.
Knowing this, you can correctly factorize polynomials and find all possible zeros.
Polynomial Division
Polynomial division is a method used to divide one polynomial by another. It's akin to long division with numbers.
In our example, we carry out polynomial division to divide our original polynomial \(g(x) = 4x^3 + 23x^2 + 34x - 10\) by the quadratic factor obtained from our complex conjugate pair, \(x^2 + 6x + 10\).
Here鈥檚 why polynomial division is essential:
  • It helps in simplifying the original polynomial into smaller factors.
  • It allows you to isolate additional factors and hence, uncover additional zeros of the polynomial.
What you want to do is arrange the terms, so you repeatedly subtract the product of the divisor and the current part of the quotient, systematically reducing the polynomial.
This process will reveal the remaining factor \(4x - 1\) in our example, showing us the path to finding all the roots.
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give back the original polynomial.
In the context of our example, once we know two factors involving complex conjugates are \(x + 3 - i\) and \(x + 3 + i\), we can multiply them to simplify them into a quadratic polynomial, \(x^2 + 6x + 10\).
Here鈥檚 why factorization is key:
  • It transforms a potentially complicated polynomial into manageable smaller factors.
  • It uncovers the roots or zeros of the polynomial giving clear insight into the behavior of the function.
By using polynomial division, we factor the original polynomial completely, revealing as its factors a quadratic \(x^2 + 6x + 10\) and a linear \(4x - 1\).
This approach lets us solve for all zeros, revealing essential characteristics of the polynomial.

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