Cubic Polynomial

A cubic polynomial is a polynomial of degree 3, meaning the highest power of the variable in the polynomial is 3. The general form of a cubic polynomial is:

f(x) = ax³ + bx² + cx + d

Where a, b, c, and d are constants, and a ≠ 0. The graph of a cubic polynomial is a curve that can have up to two turns, and it may go both above and below the x-axis.

Key Concepts

Leading Coefficient: The coefficient "a" in the term ax³ is called the leading coefficient of the cubic polynomial. It determines the end behavior of the graph.
Roots/Zeros: The roots or zeros of a cubic polynomial are the values of x for which the polynomial equals zero. A cubic polynomial can have up to 3 real or complex roots.
Turning Points: A cubic polynomial can have up to 2 turning points, where the graph changes direction.
End Behavior: The end behavior of the graph of a cubic polynomial is determined by the leading coefficient "a". If "a" is positive, the graph rises to the left and rises to the right. If "a" is negative, the graph falls to the left and falls to the right.

How to Solve and Graph Cubic Polynomials

To solve and graph a cubic polynomial, you can follow these steps:

Find the Roots: Set the polynomial equal to zero and solve for x to find the roots of the cubic polynomial. You can use factoring, the quadratic formula, or synthetic division to find the roots.
Identify Turning Points: Determine the turning points of the cubic polynomial by finding the x-coordinate of the vertex, where the derivative of the polynomial is zero.
Plot Points and Sketch Curve: Use the roots and turning points to plot points on the graph and sketch the curve of the cubic polynomial, showing its behavior as it approaches the x-axis.

Example

Let's consider the cubic polynomial f(x) = 2x³ - 3x² - 11x + 6. We can follow the steps above to solve and graph the polynomial.

Step 1: Find the Roots

Setting the polynomial equal to zero, we can use factoring or the quadratic formula to find the roots of the polynomial.

Step 2: Identify Turning Points

By finding the x-coordinate of the vertex, we can determine the turning points of the polynomial.

Step 3: Plot Points and Sketch Curve

Using the roots and turning points, we can plot points on the graph and sketch the curve of the polynomial to show its behavior.

◂Math Worksheets and Study Guides Sixth Grade. Simple Algebra

The resources above cover the following skills:

Algebra (NCTM)

Represent and analyze mathematical situations and structures using algebraic symbols.

Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.

Grade 6 Curriculum Focal Points (NCTM)

Algebra: Writing, interpreting, and using mathematical expressions and equations

Students write mathematical expressions and equations that correspond to given situations, they evaluate expressions, and they use expressions and formulas to solve problems. They understand that variables represent numbers whose exact values are not yet specified, and they use variables appropriately. Students understand that expressions in different forms can be equivalent, and they can rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information). Students know that the solutions of an equation are the values of the variables that make the equation true. They solve simple one-step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation. They construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and they use equations to describe simple relationships (such as 3x = y) shown in a table.