In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. f(x) = 2x^4 + 3x^3 + 3x - 2

Verified step by step guidance

1. Step 1: Use the Rational Root Theorem to list all possible rational roots of the polynomial. The possible rational roots are the factors of the constant term (-2) divided by the factors of the leading coefficient (2).
2. Step 2: Test each possible rational root by substituting them into the polynomial to see if they yield zero. This can be done using synthetic division or direct substitution.
3. Step 3: Once a root is found, use synthetic division to divide the polynomial by the corresponding linear factor (x - root). This will reduce the polynomial's degree.
4. Step 4: Repeat the process with the reduced polynomial to find other roots. Continue until the polynomial is factored completely into linear factors.
5. Step 5: Write the original polynomial as a product of its linear factors using the roots found in the previous steps.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n is not zero. Understanding polynomial functions is crucial for identifying their properties, such as degree, leading coefficient, and behavior at infinity.

Finding Zeros

The zeros of a polynomial function are the values of x for which f(x) = 0. These points are critical as they indicate where the graph of the polynomial intersects the x-axis. Techniques for finding zeros include factoring, using the Rational Root Theorem, synthetic division, and applying the quadratic formula for lower-degree polynomials.

Factoring Polynomials

Factoring a polynomial involves expressing it as a product of simpler polynomials, typically linear factors. This process is essential for solving polynomial equations and can be achieved through methods such as grouping, using the difference of squares, or applying the quadratic formula. Writing a polynomial as a product of linear factors reveals its zeros and provides insight into its graph.