Problem 30 - College Algebra

Question

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 obtaining the first zero or the first root. $$ f(x)=x^{3}+12 x^{2}+21 x+10 $$

Step-by-Step Solution

Answer

The roots of the function \(f(x)=x^{3}+12 x^{2}+21 x+10\) are \( x = -1, x = -2, x=-5 \).

Step 1: Rational Zero Theorem

The Rational Zero Theorem states that if a polynomial has an integer coefficient and if \( p/q \) is a zero of this polynomial, where \( p \) and \( q \) are coprime, then \( p \) is a factor of the constant term (in this case, 10) and \( q \) is a factor of the leading coefficient (in this case, 1). Therefore, for the function \( f(x)=x^{3} +12 x^{2} +21 x +10 \), the possible rationals zeros are \(\pm 1\), \(\pm 2\), \(\pm 5\), \(\pm 10\).

Step 2: Descartes's Rule of Signs

Descartes's Rule of Signs is used to determine the possible number of positive and negative real zeros in a polynomial function. By counting the sign changes in \( f(x) \) and \( f(-x) \) we can estimate the number of positive and negative roots, respectively. For function \( f(x) = x^{3} +12 x^{2} +21 x +10 \), there are no sign changes, therefore no positive real roots. For \(f(-x) = -x^{3} +12 x^{2} -21 x +10\), there are 2 sign changes, thus there are 2 or 0 negative real roots.

Step 3: Finding the roots

Using the potential rational roots from the Rational Zero Theorem, try each possible root by substituting into the polynomial until a value of zero for \( f(x) \) is found. With a trial and error approach, the roots of the polynomial are found as \( x = -1, x = -2, x=-5 \).

Step 4: Confirming with a Graph

The potential roots are now confirmed with graphing. By either using a graphing calculator, graphing software, or an online graphing calculator, the x-intercepts of the graph of the function \(f(x)=x^{3}+12 x^{2}+21 x+10\) are detected, thereby confirming the previously determined roots.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem is a powerful tool for finding potential zeros (or roots) of a polynomial. It's especially useful when dealing with polynomials with integer coefficients.
In simple terms, this theorem gives us possible candidates for rational zeros based on the factors of the constant and leading coefficients of the polynomial.
For a polynomial like \( f(x) = x^3 + 12x^2 + 21x + 10 \):

• The constant term is 10, so its factors are \( \pm 1, \pm 2, \pm 5, \pm 10 \).
• The leading coefficient is 1, so its factor is \( \pm 1 \).

From here, our possible rational zeros are all combinations \( p/q \) (where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient). This means we look at \( \pm 1, \pm 2, \pm 5, \pm 10 \) for potential zeros.
Testing these candidates by substituting them into the polynomial will help identify which, if any, achieve a zero result. This systemized approach serves as a starting point for zeroing in on viable solutions.

Descartes's Rule of Signs

Descartes's Rule of Signs offers a way to predict the number of positive and negative real roots of a polynomial by observing changes in sign.
The method involves counting how many times the coefficients of the polynomial change from positive to negative (or vice versa). This count tells us the potential number of positive real roots.
Using this rule for our polynomial \( f(x) = x^3 + 12x^2 + 21x + 10 \):

• There are no sign changes in the coefficients of \( f(x) \), indicating no positive real roots.

For negative roots, evaluate \( f(-x) \) by substituting \(-x\) into the polynomial:

• \( f(-x) = -x^3 + 12x^2 - 21x + 10 \)
• This expression shows two sign changes (from negative to positive and from positive to negative), suggesting 2 or 0 possible negative real roots.

By understanding Descartes's Rule, students can effectively narrow down which potential rational zeros are more likely to be actual roots.

Graphing Polynomial Functions

Graphing polynomial functions can be a very insightful way to visualize and confirm solutions found through algebraic methods. When you plot the graph of a polynomial, the zeros (roots) are the x-values where the graph crosses the x-axis, known as x-intercepts.
For the example polynomial \( f(x) = x^3 + 12x^2 + 21x + 10 \):

• After algebraically finding potential zeros using the Rational Zero Theorem and narrowing down possibilities with Descartes's Rule of Signs, graphing can provide a visual confirmation.
• Using a graphing calculator or graphing tool, the expected roots \(-1, -2, \text{and} -5\) should plot as points where the graph intersects the x-axis.

Graphing not only confirms these finds but also gives a broader picture of the polynomial function's behavior, showing aspects like turning points and end behavior. This approach combines analytic reasoning with visual evidence, reinforcing understanding and confidence in the solutions obtained.