Problem 44 - Algebra and Trigonometry

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 obtaining the first zero or the first root. $$ f(x)=x^{4}-4 x^{3}-x^{2}+14 x+10 $$

Step-by-Step Solution

Answer

The roots of the polynomial \(x^{4}-4 x^{3}-x^{2}+14 x+10\) are still to be confirmed as the synthetic division continues in the steps, but we have found that 2 is one of the roots.

Step 1: List possible roots using Rational Zero Theorem

The Rational Zero Theorem states that if the polynomial has integer coefficients, then every rational zero will have the form \(± p / q \), where p is a factor of the constant term and q is a factor of the leading coefficient. In the given equation, constant term is 10 and leading coefficient is 1. Therefore, possible rational roots = ± factors of 10 = ±1, ±2, ±5, ±10

Step 2: Synthetic division to find first root

Apply synthetic division to test the possible rational roots until a root is found. Synthetic division by a number, say a, is performed by writing the coefficients of the polynomial, bringing down the leading coefficient, multiplying a with the brought down number, adding the numbers vertically and again multiplying a with the new number. Continue this process until we get 0 at the end which would mean that a is a root. If we get 0 at the end, the constant above the line is the remainder, the next number is the coefficient of x in the quotient, the next is coefficient of x^2 in the quotient and so on. For this problem start testing with the smallest number, ±1. From \(f(1) = -3\) and \(f(-1) = 18\), we see that neither 1 nor -1 are roots. Continue testing and we find \(f(2) = 0\), so 2 is a root.

Step 3: Synthetic division to find remaining roots

Since 2 is a root of the polynomial, by synthetic division, the equation can be reduced to a cubic equation with coefficients obtained from synthetic division. Reduce this cubic equation by synthetic division again by trying the rational roots until another root is found. If at first we don't find any roots, we can repeat the process until we get all roots.

Step 4: Write down the result

Finally, write down the roots of the polynomial that are found.

Key Concepts

Polynomial Function

A polynomial function is an expression of several terms consisting of variables and coefficients, combined using addition, subtraction, and multiplication. In the expression \( f(x) = x^4 - 4x^3 - x^2 + 14x + 10 \), you see a polynomial of degree 4. The term with the highest power of \( x \), here \( x^4 \), determines the degree of the polynomial.

Polynomials have:

• Coefficients: Numbers multiplying the variables (e.g., \( -4 \) and \( 14 \) in \( x^3 \) and \( x \))
• Constants: Terms without variables (e.g., 10)
• Variables: The letters representing unknowns (e.g., \( x \))
• Terms: Components separated by addition or subtraction (e.g., \( x^4 \), \(-4x^3 \))

Understanding these elements is crucial since they help determine the nature of a function. Polynomials can have different features based on their degree, such as the number of possible real zeros or turning points.

Descartes's Rule of Signs

Descartes's Rule of Signs is a handy tool to predict the number of positive and negative real roots of a polynomial. Here's how it works:

For positive roots:

• Count the number of sign changes between consecutive non-zero coefficients in the polynomial.
• The number of positive real roots is equal to this number of sign changes or less by an even number.

For negative roots:

• Replace \( x \) with \( -x \) in the polynomial and count the sign changes again.
• The number of negative real roots is equal to the sign changes in this new polynomial or less by an even number.

In our polynomial, \( f(x) = x^4 - 4x^3 - x^2 + 14x + 10 \):

• Positive signs change three times: from \( +x^4 \) to \( -4x^3 \), then to \( -x^2 \), and finally to \( +14x \), indicating up to 3 positive roots or 1.
• When \( x \) is replaced with \( -x \), you'll count for negative roots. This calculation helps ascertain potential zeroes, crucial steps in using trial methods like synthetic division.

Synthetic Division

Synthetic division is a simplified form of division used to divide polynomials. It's especially useful for testing possible roots derived from the Rational Zero Theorem.

Here's how you perform synthetic division:

• Write down the coefficients of the polynomial.
• Choose a possible root, say \( a \), and place it left of the line.
• Bring down the leading coefficient to below the line.
• Multiply \( a \) by the number below the line and place the result under the next coefficient.
• Add the numbers in each column vertically, write the result below, and repeat the process until completion.

Key Outcomes:

• If the remainder is zero, \( a \) is a root of the polynomial.
• The resulting numbers below the line provide coefficients for the quotient polynomial.

For example, using the polynomial \( f(x) = x^4 - 4x^3 - x^2 + 14x + 10 \), we find that when we test the rational root 2, synthetic division confirms it as a root since it results in no remainder. This reduced the polynomial to a cubic equation, making further root-finding easier.