What are all the possible rational zeros for f(x) = 2x4 + x3 – 35x2 – 113x + 65?

In mathematics, the Rational Root Theorem is used to identify the potential rational roots of a polynomial function, particularly when the function is not factorable. These rational roots are also known as x-intercepts, zeros, and solutions. Set the polynomial to 0 to solve for them, as y=0 on our x-axis. To utilize the Rational Root Theorem, you must first grasp the fundamentals of a polynomial function. A polynomial function is one that involves only positive integer powers of x with terms that mix addition and subtraction. If the terms of a polynomial are ordered from highest exponent to lowest, it is said to be in standard form. 

Rational Zero Theorem

The rational zero theorem states that each rational zero(s) of a polynomial with integer coefficients is of the form p/q,

f(x) = a_nxⁿ+a_n-1x^n-1+….+a₂x²+a₁x¹+a₀

Where,

• p is a factor of the constant a₀ and 
• q is a factor of the leading coefficient a_n.
• p and q are relatively prime 

A rational root of a polynomial function,

f(x) = a_nxⁿ+a_n-1x^n-1+….+a₂x²+a₁x¹+a₀

p/q = a factor of last term (a₀)/a factor of first term (a_n).

Note: a_n, a_{n – 1}, …, a₂, a₁, a₀ are integers.

As an example, 3x³ + 2 – 4x⁵ in standard form is -4x⁵ + 3x³ + 2. If the polynomial is in standard form, the leading coefficient is the first term; otherwise, turn to the term with the highest exponent. A polynomial’s constant term is the sole term that does not have a variable. In the previous example, the leading coefficient is -4, and the constant is 2. Finally, a rational number is one that can be written as the quotient or fraction of two integers. This indicates that numbers like are not logical since they include repeated decimals. 

According to the Rational Root Theorem, a rational solution for a polynomial equation with integer coefficients must be represented by a factor of the constant (denoted by the variable p) divided by a factor of the leading coefficient (denoted by the variable q). This means that the rational root theorem finds all potential solutions to a polynomial by listing all of the components of the leading coefficient and constant and displaying them as a fraction. This is represented by the following list of potential roots: ±p/q

The “rational zero theorem” (or “rational root theorem”) is another name for the rational zero tests. It is used to generate a list of polynomial function’s potential rational zeros. The rational root theorem is used to determine the set of all possible rational zeros of a polynomial function (or) the rational roots (solutions) of a polynomial equation.

How Does the Rational Zero Theorem Help You Find Possible Rational Zeros?

To calculate the rational zeros of a polynomial function f (x),

• Find the constant and its components. Each number symbolizes the letter p.
• Determine the leading coefficient and its components. Each number symbolizes the letter q.
• Find all feasible p/q pairings, and these are all possible rational zeros.
• All of these may not be the true origins. Simply replace to discover which of the rational zeros satisfies f(x) = 0.

How Does the Rational Zero Theorem Work?

The rational zero theorem is another name for the rational zero theorem. Here are some basic measures to take:

• Arrange f(x) in descending order of variable exponents.
• Write the constant term’s factors, i.e. all the possible values of p.
• Write down the factors of the leading coefficient, or all of the possible values of q.
• Write out all of the potential p/q values. Because variables might be negative, both p/q and – p/q must be provided. Remove any duplication and simplify each value.
• To find the values of p/q for which f(p/q) = 0, use synthetic division. All of these are the logical roots of f. (x).

Rational Root System

If a polynomial function includes integer coefficients and is stated in decreasing order of exponents, then every rational zero must be of the form ± p/ q, where p is a constant term factor and q is a leading coefficient factor. Rational Zero: A value x ∈ Q such that f(x)=0. In other words, x is a rational number that when input into the function f, the output is 0. If a polynomial has integer coefficients, then all of its zeros will be of the form p/q where p is a factor of the constant term and q is a factor of the coefficient of the leading term.

Steps for Using the Rational Zeros Theorem to Find All Possible Rational Zeros With Repeated Possible Zeros :

Step 1: Find all factors (p) of the constant term. Factors can be negative so list ± for each factor.

Step 2: Find all factors (q)of the leading term. (The term that has the highest power of x).

Step 3: List all possible combinations of ±p/q as the possible zeros of the polynomial. Simplify the list to remove and repeated elements.

What are all the possible rational zeros for f(x) = 2x⁴ + x³ – 35x² – 113x + 65?

Solution:

f(x) = 2x⁴ + x³ – 35x² – 113x + 65

Step 1: Find all factors (p) of the constant term. We are looking for the factors of 65, which are ±1, ±5, ±13, and ±65.

Step 2: Find all factors (q)of the coefficient of the leading term. We are looking for the factors of 2, which are ±1, and ±2.

Step 3:  List all possible combinations of ±p/q as the possible zeros of the polynomial. Simplify the list to remove and repeated elements. This means we have, p/q = ±1, ±5, ±13, ±65/±1, ±2 which gives us the following list, 

±1/1, ±5/1, ±13/1, ±65/1, ±1/2, ±5/2, ±13/2, ±65/2.

Now, we simplify the list and eliminate any duplicates. Thus the possible rational zeros of the polynomial are:

±1, ±5, ±13, ±65, ±1/2, ±5/2, ±13/2, ±65/2.

Step 4: Using synthetic division,

2x⁴ + x³ – 35x² – 113x + 65/(x – 5)

= 2x³ + 11x² + 20x – 13

Again dividing 2x³ + 11x² + 20x – 13/(x – 1/2)

= 2x²+12x+26

Since the remainder is 0, x – 5 and x-1/2 are factors. the other factor is the quotient :

2x² + 12x + 26 = 2(x² + 6x + 13)

Using the quadratic formula to find the zeros of x² + 6x + 13 = 0;

By solving the Quadratic formula we will get the value: -3 ± 2i.

The complex zeros for the given equation are 5, 1/2, -3 – 2i, -3 + 2i.                                                                        

Conceptual Questions                                           

Question 1: What is the Definition of the Rational Root Theorem?

Answer:

According to the rational root theorem, a polynomial’s rational zero has the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Question 2: Why is the Rational Root Theorem Important?

Answer: 

The rational root theorem is used to determine the set of all possible rational zeros of a polynomial function (or) the rational roots (solutions) of a polynomial equation.

Question 3: What do p and q in the Rational Zero Theorem stand for?

Answer: 

p and q represent all potential rational roots of a polynomial in the rational zero theorem. p represents all positive and negative factors of the polynomial’s constant, whereas q represents all positive and negative factors of the polynomial’s leading coefficient.

Question 4: Is there a difference between rational roots and rational zeros?

Answer: 

Rational roots are also referred to as rational zeros. Finding the rational x-intercepts of a polynomial is regarded the same.

Sample Questions

Question 1: Find the possible rational roots of x³ – x² – 10x – 8 = 0. Then determine the rational roots?

Solution: 

According to the Rational Root Theorem, if p/q is the root of the equation then p is the factor of 8 and q is the factor of 1

Possible values of p: ±1, ±2, ±4, ±8

Possible values of q: ±1

Possible rational roots, p/q: ±1, ±2, ±4, ±8

Thus, x³ – x² – 10x – 8 = (x + 2)(x² – 3x – 4). Factoring x²– 3x – 4 yields (x – 4)(x + 1). The roots of x³– x²-10x – 8 = 0 are -2, -1, and 4.

Question 2: Find the possible rational roots of y = x⁵ + 3x² − 10x – 24.

Solution:

We need all of the available factors, both positive and negative, from our leading and lagging coefficients to employ the Rational Root Theorem. Our key factors are: ±1. The last term’s components are more complicated: ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.

What a slog. Now we take the second set (from the constant term) and layer it on top of the first set. It may appear to be backwards, but that is how we do it. Overhead term is constant.

That is simple in our instance since the leading phrases are  ±1. As a result, our final list of viable variables is as follows:  ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.

Question 3: Find the possible rational roots of y = 3x⁷−12x³ + 52x² − 96x + 9.

Solution: 

The constant factors are ±1, ±3, ±9. These are the ones that go on top. The leading factors are ±1, ±3. We’ll stick these below. Let’s write all the combinations out: 1/1, 1/3, 3/1, 3/3, 9/1, 9/3. We can have any combination of positive and negative numbers our hearts desire, so these are all ± as well. 

The final answer is ±1, ±3, ±9, and ±1/3.