Problem 39 - 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. $$ 4 x^{4}-x^{3}+5 x^{2}-2 x-6=0 $$

Step-by-Step Solution

Answer

The zeros of the polynomial are x = -1, 1/4, 1/2, 3

Step 1: Application of Rational Zero Theorem

According to the Rational Zero Theorem, if the polynomial \(4x^{4}-x^{3}+5x^{2}-2x-6=0\) has a rational zero \(p/q\), then \(p\) is a factor of the constant term, -6, and \(q\) is a factor of the coefficient of the leading term, 4. Possible rational zeros include \(±1, ±2, ±3, ±6, ±1/2, ±1/4\)

Step 2: Use Descartes's Rule of Signs to check the number of possible positive and negative zeros

Descartes's rule of signs states that the number of positive real roots of a polynomial is equal to the number of changes in sign between terms, or less than that by a multiple of 2. In our equation, the terms switch signs three times, which means the equation has either three or one positive real roots. The number of negative real roots of a polynomial is the number of sign changes after replacing \(x\) with \(-x\) in the polynomial, or less than that number by a multiple of 2. After substituting \(x\) with \(-x\) our equation turns into: \(4x^{4}+x^{3}+5x^{2}+2x-6\), and we can see that terms switch signs one time, which infers there is one negative real root.

Step 3: Test each possible Rational Zero

We substitute each possible rational zero into the given polynomial. If the result equals zero, then it is a root of the equation. So the possible rational roots that would satisfy the equation after testing are -1 , 1/2 and 1/4.

Step 4: Factor the polynomial

Factorizing the polynomial using the roots obtained, we get \((x+1)(4x-1)(4x-2)(x-3)=0\)

Step 5: Solving for x

Equating each factor to zero gives us more roots, \(x = -1, 1/4, 1/2, 3\)

Key Concepts

Rational Zero Theorem

Understanding the Rational Zero Theorem is crucial for solving polynomial equations. It provides a systematic way to find all possible rational zeros, or roots, of a polynomial equation. According to the theorem, if a polynomial has rational roots in the form of \frac{p}{q}, where both p and q are integers, then p is a factor of the constant term, and q is a factor of the leading coefficient.

For example, consider the polynomial equation:
$$ 4x^{4} - x^{3} + 5x^{2} - 2x - 6 = 0 $$.
The constant term here is -6, and potential factors of -6 are ±1, ±2, ±3, and ±6, while the leading coefficient is 4, whose factors are ±1, ±2, and ±4. Therefore, the possible rational zeros include ±1, ±1/2, ±1/4, ±2, ±3, and ±6. Once you have the list of possible zeros, the next step is to identify which, if any, are actual zeros of the polynomial by substitution.

Descartes's Rule of Signs

Descartes's Rule of Signs empowers students to predict the number of positive and negative real zeros in a polynomial equation. This rule states that the number of positive real zeros is either the number of times the coefficients of the terms change signs or less than that by a multiple of 2. Similarly, for negative real zeros, first replace every occurrence of x with -x and then apply the sign change count.

For instance, if we consider the equation from our example:
$$ 4x^{4} - x^{3} + 5x^{2} - 2x - 6 = 0 $$
,we note there are three changes in sign, suggesting there could be three or one positive real zeros. For negative zeros, after substituting x with -x, we have:$$ 4x^{4} + x^{3} + 5x^{2} + 2x - 6 $$
,indicating just one change in sign, hence predicting one negative real zero. This rule doesn't give the exact zeros but limits the possibilities and guides further exploration.

Polynomial Roots

The roots (or zeros) of a polynomial equation are the values of x for which the polynomial evaluates to zero. These values are critical as they are the points where the graph of the polynomial crosses or touches the x-axis. Polynomial equations can have real and complex roots, and the highest degree of the polynomial indicates the maximum number of roots.

For example, in the equation:
$$ 4x^{4} - x^{3} + 5x^{2} - 2x - 6 = 0 $$
,the polynomial is of degree 4, indicating up to 4 roots. Root-finding is a blend of numerical methods, graphing utilities, and algebraic techniques like factoring, synthetic division, or using theorems such as Rational Zero and Descartes's Rule. Sometimes, one or more roots can be complex, and they always occur in conjugate pairs. The actual roots can be rational, irrational, or complex numbers.

Factoring Polynomials

Factoring is a critical algebraic process used in finding the roots of polynomials. When a polynomial is factored completely, it is expressed as a product of its factors, which may include constants, linear expressions, and irreducible quadratic expressions. If the polynomial can be factored over the set of integers or rational numbers, the factors might reveal the roots directly.

In our earlier example:
$$ 4x^{4} - x^{3} + 5x^{2} - 2x - 6 = 0 $$
,after finding the roots using the Rational Zero Theorem and Descartes’s Rule of Signs, the polynomial can be factored as:
$$ (x+1)(4x-1)(4x-2)(x-3) = 0 $$
,Each factor set equal to zero yields a root of the original polynomial. The factoring step is essential as it transforms the polynomial equation into a set of simpler equations, making the solutions more accessible.