Question

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=2 x^{4}-x^{3}-5 x^{2}+2 x+2 $$

Short answer

The zeros are 1, -1/2, √2, and -√2. The factored form is \( (x - 1)(2x + 1)(x - √2)(x + √2) \).

Step-by-step solution

Identify the polynomial and leading coefficients

Given the polynomial function: f(x) = 2x^4 - x^3 - 5x^2 + 2x + 2, identify the leading coefficient and the constant term. The leading coefficient (a) is 2 and the constant term (b) is 2.

List all possible rational zeros

According to the Rational Zeros Theorem, the possible rational zeros are the ratios of the factors of the constant term (b) to the factors of the leading coefficient (a). Therefore, Possible rational zeros = \(\frac{\text{factors of } b}{\text{factors of } a}\) = \(\frac{\pm 1, \pm 2}{\pm 1, \pm 2}\) = \(\frac{\pm 1, \pm 2}{\pm 1, \pm 2}\). This gives the possible rational zeros as ±1, ±2, ±1/2.

Test the possible rational zeros using synthetic division

Use synthetic division to test each possible rational zero from the list. Start with 1:1 | 2 -1 -5 2 2--| ----------------__| 2 1 -4 -2 0Since the remainder is 0, x = 1 is a zero of the polynomial.

Factor out the zero and simplify the polynomial

Since x = 1 is a zero, factor the polynomial as (x - 1) and divide the polynomial by (x - 1) using synthetic division:1 | 2 -1 -5 2 2--| ----------------__| 2 1 -4 -2 0Thus, the quotient is 2x^3 + x^2 - 4x - 2. Now factor 2x^3 + x^2 - 4x - 2.

Factor the remaining polynomial

Use the Rational Zeros Theorem again on the remaining polynomial 2x^3 + x^2 - 4x - 2. Repeat the process to find additional zeros. Testing -1/2 using synthetic division:-1/2 | 2 1 -4 -2----| ---------------- | 2 0 -4 0Since the remainder is 0, x = -1/2 is a zero.

Factor out the zero and simplify further

Since x = -1/2 is a zero, factor it out as (2x + 1) and perform division:2x^3 + x^2 - 4x - 2 = (x - 1)(2x + 1)(x^2 - 2)Now factor x^2 - 2.

Express the polynomial in factored form

The quadratic x^2 - 2 can be factored into (x - √2)(x + √2). Thus, the polynomial function f(x) is factored as: f(x) = (x - 1)(2x + 1)(x - √2)(x + √2).

Key concepts

Polynomial Functions

Polynomial functions are equations that involve terms with varying powers of a variable, typically denoted as x. These functions are expressed as sums where each term is the product of a constant coefficient and the variable raised to a non-negative integer power. For instance, in the polynomial function given in the exercise, 2x^4 - x^3 - 5x^2 + 2x + 2, each term such as 2x^4 or -x^3 corresponds to a combination of coefficients and powers of x. Understanding the structure of polynomials, including terms, degrees (the highest power of x), and coefficients, is crucial. Polynomials can be quadratic, cubic, quartic, etc., based on their highest degree, and the zeros (or roots) of these functions are the values of x for which the polynomial equals zero.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form (x - c). It is particularly useful in finding roots or zeros of polynomial functions. Here’s how it’s done:

• Write down the coefficients of the polynomial. If any degrees of x are missing, use a zero for that coefficient.
• Place the possible zero to test (let's say c) to the left, separating it with a vertical bar.
• Bring down the leading coefficient.
• Multiply this leading coefficient by c, place the result under the next coefficient, and add downward.
• Repeat the multiplication and addition until no coefficients remain.

A zero remainder indicates that the value tested is indeed a zero of the polynomial. For example, 1 is a zero for the polynomial 2x^4 - x^3 - 5x^2 + 2x + 2 as shown in our step-by-step solution. Synthetic division efficiently simplifies polynomials and helps in factoring.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler, multiplicative components, where each component is a factor of the original polynomial. Utilizing synthetic division to find zeros, you can then factor the polynomial incrementally. For the given polynomial 2x^4 - x^3 - 5x^2 + 2x + 2, we found two zeros via synthetic division, x = 1 and x = -1/2. This allows the initial polynomial to be factored into (x - 1) and (2x + 1) while leaving a quotient polynomial that can be simplified further.

• Identify and test possible rational zeros.
• Use synthetic division to verify each zero.
• Factor the polynomial with the verified zeros step by step.

The remaining quadratic x^2 - 2 can be factored using quadratic techniques into (x - √2)(x + √2), completing the factorization of the polynomial into (x - 1)(2x + 1)(x - √2)(x + √2).

Zeros of Functions

The zeros of a polynomial function are the values of x that make the function equal to zero. These values are also referred to as roots or solutions of the polynomial equation. Finding zeros is essential because they help in the factorization process and provide critical points for graphing the polynomial function. To locate zeros using the Rational Zeros Theorem, calculate the potential rational zeros as ratios of the factors of the constant term to the factors of the leading coefficient. Each value is then tested using methods like synthetic division.

• For a polynomial 2x^4 - x^3 - 5x^2 + 2x + 2, possible rational zeros derive from the theorem as ±1, ±2, ±1/2.
• Synthetic division confirms zeros, simplifying polynomial factorization.
• Zeros such as x = 1 and x = -1/2 guide the factorization into fractions like (x - 1)(2x + 1)(x - √2)(x + √2).

Identifying these zeros correctly aids in fully understanding polynomial behavior and solving related algebraic problems.