Problem 23 - Algebra 2

Question

Find all of the zeros of each function. \(g(x)=x^{3}+6 x^{2}+21 x+26\)

Step-by-Step Solution

Answer

The zeros are \( x = -3 \pm i\sqrt{15} \) and \( x = -1 \).

Step 1: Establish the Polynomial Equation

We are given the function \( g(x) = x^3 + 6x^2 + 21x + 26 \). Our task is to find the zeros of this cubic polynomial, which are the values of \( x \) that satisfy \( g(x) = 0 \).

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational solution of the polynomial equation is a factor of the constant term divided by a factor of the leading coefficient. Here, the possible rational roots could be \(\pm 1, \pm 2, \pm 13, \pm 26\).

Step 3: Perform Synthetic Division or Subsition for Possible Roots

Test the possible roots by substituting them into \( g(x) \) or using synthetic division to determine if any equates to zero. Testing these values:- For \( x = -1 \), calculate \( g(-1) = (-1)^3 + 6(-1)^2 + 21(-1) + 26 = -1 + 6 - 21 + 26 = 10 \) (not zero). Continue testing each prospect. None work for \( x = -1, 1, 2, -2, 13, -13, 26, -26 \).

Step 4: Use Numerical or Graphical Methods or Polynomial Solver

Since no rational roots work, we turn to numerical methods or graphing calculators. Upon using a solver or graphing, we find that the roots include complex numbers since they cannot be factored using the Rational Root Theorem alone.

Step 5: Confirm the Complex Roots

Using a cubic polynomial calculator or software, it is confirmed that the roots are \( x = -3 + i\sqrt{15} \), \( x = -3 - i\sqrt{15} \), and \( x = -1 \). Here, \( i \) is the imaginary unit.

Key Concepts

Zeros of Polynomial

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. For any polynomial function, finding these zeros helps us understand the behavior of the graph of the function. For instance, the polynomial \(g(x) = x^3 + 6x^2 + 21x + 26\) is a cubic polynomial, meaning it can have up to three real or complex zeros. Identifying these zeros is crucial as it shows where the function crosses or touches the x-axis on a graph, giving insights into the specific turning points and intercepts of the polynomial.

• The zeros can be real or complex numbers.
• They indicate where the function evaluates to zero.
• Zeros are often referred to as roots of the polynomial.

By graphing, using the Rational Root Theorem, or utilizing numerical methods, we strive to find and confirm these zeros.

Rational Root Theorem

The Rational Root Theorem is a valuable tool in determining possible rational zeros of a polynomial. It states that any rational zero of a polynomial with integer coefficients will be a fraction \(\frac{p}{q}\), where \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient. This theorem simplifies the process of finding rational zeros by narrowing down the possibilities.
When applied to the polynomial \(g(x) = x^3 + 6x^2 + 21x + 26\), the potential rational roots are obtained by considering the factors of the constant term (26) and the leading coefficient (1). This results in the possible rational roots: \(\pm 1, \pm 2, \pm 13, \pm 26\). Although these are the initial candidates, they need to be tested further to confirm if they satisfy the polynomial equation.

Complex Numbers

Complex numbers come into play when a polynomial has no real zeros. They are numbers of the form \(a + bi\) where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined as \(\sqrt{-1}\).

• Complex roots appear in conjugate pairs; for example, if \(a + bi\) is a root, \(a - bi\) is also a root.

In our polynomial \(g(x) = x^3 + 6x^2 + 21x + 26\), after exhausting rational possibilities and using numerical methods, we find complex roots: \(-3 + i\sqrt{15}\) and \(-3 - i\sqrt{15}\). These complex roots reflect the polynomial's behavior beyond real numbers, indicating more intricate relationships between the polynomial's values.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form \(x - c\). It's especially useful for testing potential roots determined by the Rational Root Theorem. By using synthetic division, we can determine if a suspected root is an actual root by checking if the remainder is zero.
To perform synthetic division on our polynomial \(g(x)\) with potential root candidates, we set up a synthetic division table with the coefficients: 1, 6, 21, and 26 corresponding to \(x^3, x^2, x,\) and the constant terms. If dividing by a trial root results in zero remainder, it confirms that the trial value is indeed a root.
Despite rigorous testing using synthetic division, none of the rational candidates we tested for \(g(x)\) led to zero, thus confirming that the polynomial only has complex roots or requires deeper investigation through alternative numeric methods.