Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x⁴+x³-6x²-7x-2

Verified step by step guidance

1. Start by writing down the polynomial function: \(f(x) = 2x^4 + x^3 - 6x^2 - 7x - 2\).
2. Attempt to factor the polynomial by grouping or using the Rational Root Theorem to find possible roots. The Rational Root Theorem suggests testing factors of the constant term (\(\pm1, \pm2\)) over factors of the leading coefficient (\(\pm1, \pm2\)).
3. Test possible rational roots by substituting them into \(f(x)\) to see if they yield zero. Each root found corresponds to a factor of the form \((x - r)\).
4. Once a root is found, use polynomial division (long division or synthetic division) to divide \(f(x)\) by the corresponding factor to reduce the polynomial to a cubic or quadratic.
5. Continue factoring the reduced polynomial until it is completely factored into linear and/or irreducible quadratic factors. Then, use these factors to sketch the graph by identifying zeros (roots), their multiplicities, and the end behavior based on the leading term.

Key Concepts

Polynomial Functions and Their Graphs

A polynomial function is an expression consisting of variables raised to whole-number exponents and coefficients. Understanding the general shape of polynomial graphs, including end behavior and turning points, helps in sketching the graph accurately. The degree of the polynomial determines the maximum number of turning points and the end behavior of the graph.

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or factors. This process is essential for finding the roots or zeros of the polynomial, which correspond to the x-intercepts of its graph. Factoring techniques include grouping, synthetic division, and using the Rational Root Theorem.

Finding Zeros and Their Multiplicities

The zeros of a polynomial are the values of x that make the function equal to zero. Each zero's multiplicity affects the graph's behavior at that point: odd multiplicities cross the x-axis, while even multiplicities touch and turn around. Identifying zeros and their multiplicities is crucial for accurately sketching the polynomial's graph.