Question

Find bounds on the real zeros of each polynomial function. $$ f(x)=-4 x^{5}+5 x^{3}+9 x^{2}+3 x-12 $$

Short answer

The real zeros of the polynomial function are bounded by the smallest and largest valid rational roots.

Step-by-step solution

Confirm Polynomial and Leading Coefficient

Identify the given polynomial function \[ f(x) = -4x^5 + 5x^3 + 9x^2 + 3x - 12 \] The leading coefficient is -4.

Apply the Descartes' Rule of Signs for Positive Roots

Observe the sign changes in the polynomial function for positive roots. Write down the number of sign changes. \[ f(x) = -4x^5 + 5x^3 + 9x^2 + 3x - 12 \] The sequence of coefficients is -4, 5, 9, 3, and -12, which changes signs three times: -4 to +5, +3 to -12. Therefore, there are 3 or 1 possible positive zeros.

Apply Descartes' Rule of Signs for Negative Roots

Substitute \[ x = -x \] into the polynomial and find sign changes: \[ f(-x) = -4(-x)^5 + 5(-x)^3 + 9(-x)^2 + 3(-x) - 12 \] Simplify the equation: \[ f(-x) = 4x^5 - 5x^3 + 9x^2 - 3x - 12 \] There are 4 sign changes in 4, -5, 9, -3, and -12, meaning there are 4 or 2 or 0 possible negative zeros.

Use the Rational Root Theorem

List the possible rational roots using factors of the constant term (-12) divided by factors of the leading coefficient (-4). List of possible roots: \[ \frac{\text{Factors of }\text{-12}}{\text{Factors of }\text{-4}} \rightarrow \frac{\text{-12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12}}{\text{-4, -2, -1, 1, 2, 4}} \] Possible rational roots: \[ \text{±1, ±3, ±4, ±12, ±\frac{1}{2}, ±\frac{3}{2}, ±\frac{1}{4}, ±\frac{3}{4} } \]

Test the Possible Rational Roots

Use synthetic division or direct substitution to test each rational root. If a particular root yields zero value for \(f(x)\), then that root is valid. Continue this process to identify all possible rational roots.

Establish Boundaries

After testing all possible rational roots, determine the smallest and largest valid roots to establish boundaries. This enables the identification of lower and upper bounds for real zeros within those intervals.

Key concepts

Descartes' Rule of Signs

Descartes' Rule of Signs is a useful method in determining the number of positive and negative roots in polynomial equations. The rule states that the number of positive real zeros of a polynomial is either equal to the number of sign changes in the sequence of its coefficients or less than it by an even number.
This rule applies similarly for negative roots, but with a twist. You need to replace x with -x in the polynomial and then count the sign changes.
For example, with the polynomial \(f(x) = -4x^5 + 5x^3 + 9x^2 + 3x - 12\), the sequence of coefficients is \(-4, 5, 9, 3, -12\). Counting the sign changes here, we notice three changes: from -4 to 5, from 9 to 3, and from 3 to -12. This means there are 3 or 1 positive roots.
When we replace x with -x to get \(f(-x) = 4x^5 - 5x^3 + 9x^2 - 3x - 12\), the sequence becomes \(4, -5, 9, -3, -12\). There are 4 sign changes, implying that there are 4, 2, or 0 negative roots.

Rational Root Theorem

The Rational Root Theorem helps in identifying possible rational roots of polynomial equations. According to this theorem, any rational root of a polynomial equation must be a fraction where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient.
For the polynomial \(f(x) = -4x^5 + 5x^3 + 9x^2 + 3x - 12\), the constant term is -12 and the leading coefficient is -4.
The possible rational roots are \(\pm 1, \pm 3, \pm 4, \pm 12\) and fractions such as \(\pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}\).
By testing these values using synthetic division or direct substitution, we can determine which of these values are actually roots of the polynomial equation.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number exponents and their coefficients. They take the general form \(P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\) where each \(a_i\) is a constant coefficient and \(x\) is the variable.
In the exercise, the polynomial given is \(f(x) = -4x^5 + 5x^3 + 9x^2 + 3x - 12\). Here, -4 is the coefficient of the highest degree term \(x^5\).
This form of equation can be analyzed and solved to find the roots, or values of x, that make the polynomial equal to zero. These roots can be real or complex, and understanding their nature and finding them is a key aspect of algebra.

Roots of Equations

The roots of a polynomial equation are the values of x for which the polynomial equals zero. Solving polynomial equations involves finding these roots.
They can be real or complex numbers. Real roots are the x-values where the polynomial curve intersects the x-axis.
There are different methods to find the roots of equations, such as factoring, using the Rational Root Theorem, Descartes' Rule of Signs, and synthetic division.
It is often a step-by-step process of identifying possible roots and then testing them to see if they satisfy the polynomial equation.

Synthetic Division

Synthetic division is a method used to divide polynomials and test potential roots quickly. This method simplifies the process compared to long division.
To use synthetic division to test a possible root, we set up a synthetic division table with the coefficients of the polynomial.
For example, let's test the root 1 for the polynomial \(f(x) = -4x^5 + 5x^3 + 9x^2 + 3x - 12\). We write down the coefficients \[-4, 0, 5, 9, 3, -12\] (note the zero for the missing \(x^4\) term).
We then perform synthetic division by writing the possible root outside the division box, and perform the necessary operations step by step to see if the remainder is zero.
If the remainder is zero, the tested root is indeed a solution to the polynomial. This process helps to identify all valid roots of the polynomial equation.