Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=2x^4+x^3-6x^2-7x-2

Verified step by step guidance

1. Identify the polynomial function: \( f(x) = 2x^4 + x^3 - 6x^2 - 7x - 2 \).
2. Check if the polynomial can be factored by looking for common factors or using techniques like synthetic division or the Rational Root Theorem.
3. If a factor is found, express the polynomial in its factored form. For example, if \( x - a \) is a factor, then \( f(x) = (x - a)Q(x) \) where \( Q(x) \) is the quotient.
4. Once factored, identify the roots of the polynomial by setting each factor equal to zero and solving for \( x \).
5. Use the roots and the leading coefficient to sketch the graph, noting the end behavior and the multiplicity of each root to determine the shape of the graph at each intercept.

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 of a polynomial in one variable is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 'n' is a non-negative integer and 'a_n' are constants. Understanding the degree and leading coefficient of the polynomial is crucial for graphing and analyzing its behavior.

Factoring Polynomials

Factoring polynomials involves rewriting the polynomial as a product of simpler polynomials or linear factors. This process is essential for finding the roots of the polynomial, which are the x-values where the function equals zero. Techniques such as grouping, using the quadratic formula, or applying the Rational Root Theorem can be employed to factor polynomials effectively.

Graphing Polynomial Functions

Graphing polynomial functions requires understanding their key features, such as intercepts, end behavior, and turning points. The roots found from factoring indicate where the graph crosses the x-axis, while the degree of the polynomial determines the number of turning points. Analyzing these characteristics helps in sketching an accurate graph of the polynomial function.