Problem 40 - College Algebra

Question

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ f(x)-x^{3}-4 x^{2}-7 x+10 $$

Step-by-Step Solution

Answer

The zeros of the polynomial function \(f(x)=x^{3}+12 x^{2}+21 x+10\) are -1, -2, and -10.

Step 1: The Rational Zero Theorem

List the ratio of factors of the constant term 10 to factors of the leading coefficient 1. This gives potential rational zeros: -10, -5, -2, -1, 1, 2, 5, 10.

Step 2: Descartes’s Rule of Signs

Count number of sign changes in the polynomial to find potential positive real roots. The polynomial is \(f(x)=x^{3}+12 x^{2}+21 x+10\) and has no sign changes. Therefore, it has no positive roots. By substituting -x for x, you get a polynomial \(f(-x)= -x^3+12x^2-21x+10\), which has 2 sign changes, therefore there could be 2 or 0 negative real roots.

Step 3: Finding the zeros

From step 1 and 2, try to factorize the polynomial or use synthetic division to find the roots. It can be observed that \(x+1\), \(x+2\), and \(x+10\) are factors of the polynomial which gives the roots -1, -2, -10.

Step 4: Verification using a graph

Graph the polynomial function \( f(x)=x^{3}+12 x^{2}+21 x+10 \) using a graphing utility and check if the x-intercepts are -1,-2 and -10 which will verify these found solutions.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a helpful tool in determining potential rational zeros of a polynomial. This theorem suggests that for a polynomial function with integer coefficients,

• any potential rational zero will be in the form \( \frac{p}{q} \),
• where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.

In the polynomial \( f(x) = x^3 + 12x^2 + 21x + 10 \), the constant term is 10, and the leading coefficient is 1.This simplifies our process because any factor of 10 also works as a factor for 1. Thus, the potential rational zeros are the factors of 10: -10, -5, -2, -1, 1, 2, 5, and 10. These potential zeros guide us in testing which ones might be actual roots of the polynomial.

Descartes's Rule of Signs

Descartes's Rule of Signs provides a quick way to estimate the number of positive and negative real roots a polynomial may have by checking sign changes. To use this rule:

• Examine the sign changes in the order of the polynomial function for positive roots.
• Substitute \(-x\) into the function to identify sign changes for negative roots.

In this exercise, the polynomial \( f(x) = x^3 + 12x^2 + 21x + 10 \) exhibits no sign changes, indicating zero positive real roots.Conversely, by substituting \(-x\) for \(x\), we have \( f(-x) = -x^3 + 12x^2 - 21x + 10 \), showcasing 2 sign changes.This suggests there could be 2 or 0 negative real roots.It’s an excellent tool for narrowing down the possible real zeros and sets a direction for further testing.

Synthetic Division

Synthetic division is a streamlined method for performing polynomial division, particularly useful for dividing by linear expressions of the form \(x-c\). Follow these steps:

• Write down the coefficients of the polynomial and the potential zero you want to test.
• Simplify the polynomial using these numbers to determine if the division results in a remainder of zero indicating a root.

For \(f(x) = x^3 + 12x^2 + 21x + 10\), choose roots like -1, -2, and -10 from our earlier findings.Try synthetic division with each, and if it results in a remainder of zero, it confirms the root.Ultimately, it helps pinpoint actual zeros effectively with fewer calculations compared to long division, especially useful in combination with the Rational Zero Theorem.

Polynomial Graph

Graphing a polynomial is a practical way to visualize roots and verify solutions. By plotting the function \( f(x) = x^3 + 12x^2 + 21x + 10 \) on a graphing utility:

• Focus on the x-axis intercepts, where the polynomial equals zero.
• Check if the intercepts match the zeros found through calculation, such as -1, -2, and -10 in this problem.

The visual curve helps not only in confirming the solutions but also in understanding the behavior of the polynomial.The degree of the polynomial, number of turning points, and end behavior can all be assessed.This graphical approach rounds out the analytical findings from algebraic methods, painting a complete picture of polynomial behavior.