Understanding Polynomial Graphs

Polynomials are mathematical expressions that involve variables raised to whole number powers. The graph of a polynomial function is a smooth, continuous curve. Understanding the behavior of polynomial graphs is crucial for solving problems on the SAT.

Key Characteristics of Polynomial Graphs

• Degree: The highest power of the variable in the polynomial. It determines the number of roots and the end behavior of the graph.
• Leading Coefficient: The coefficient of the term with the highest degree. It affects the direction in which the graph opens.
• Roots/Zeros: Points where the graph intersects the x-axis. The number of roots can be at most equal to the degree of the polynomial.
• End Behavior: Describes how the graph behaves as $x$ approaches positive or negative infinity.

Example Problem

Consider the polynomial function:
$f(x) = 2x^{3} - 4x^{2} - 6x + 8$

Step 1: Determine the Degree and Leading Coefficient

• The degree is 3 (the highest power of $x$).
• The leading coefficient is 2 (the coefficient of $2x^{3}$).

Step 2: Find the Roots

To find the roots, set $f(x) = 0$: $2x^{3} - 4x^{2} - 6x + 8 = 0$ Using synthetic division or the Rational Root Theorem, we can find one root: $x = 2$.

Step 3: Factor the Polynomial

We can factor using the root we found: $f(x) = 2(x - 2)(x^{2} + 2)$

Step 4: Analyze the Factorization

• The quadratic $x^{2} + 2$ has no real roots (as it doesn't intersect the x-axis), so the only real root is $x = 2$.
• The graph will touch the x-axis at $x = 2$ and turn around.

Step 5: Determine End Behavior

• As $x o ext{positive infinity}$, $f(x) o ext{positive infinity}$ (since the leading coefficient is positive).
• As $x o ext{negative infinity}$, $f(x) o ext{negative infinity}$.

Key Questions

• What is the degree of the polynomial $f(x) = 3x^{4} - 5x^{3} + x - 12$?
• How many real roots does the polynomial $g(x) = x^{2} - 5x + 6$ have?

• What does the graph of a polynomial with an odd degree and a positive leading coefficient look like?
• If a polynomial has a root at $x = -1$, what can you deduce about the factors of the polynomial?

Keywords

• Polynomial
• Degree
• Roots
• Leading Coefficient
• End Behavior