Polynomials

Polynomials are an essential part of mathematics that you will encounter often in algebra. Understanding polynomials is important because they form the basis of the equations you solve in mathematics, physics, engineering, and even computer science.

What are polynomials?

A polynomial is a mathematical expression consisting of a sum of powers in one or more variables multiplied by coefficients. The word "polynomial" comes from the Greek words "poly" (meaning "many") and "nomial" (meaning "term"), so it basically means "many terms."

A simple variant of a polynomial in one variable, usually x, is written like this:

 a n *x n + a n-1 *x n-1 + ... + a 2 *x 2 + a 1 *x + a 0

Where:

• a n, a n-1, ..., a 0 are constants called coefficients.
• n is the highest power, called the degree of the polynomial.
• x is the variable.

Understanding polynomials through examples

Example 1: A simple polynomial

Consider the polynomial:

 2x 3 + 4x 2 - x + 7

This polynomial has:

• Degree of 3 (because of 2x 3 term).
• The coefficients are 2, 4, -1, and 7.
• Four positions.

Example 2: Another polynomial

Let us consider another polynomial:

 5y 4 - 3y 2 + 2

We are here:

• The degree of 4 (from the term 5y 4).
• The coefficients are 5, 0, -3 and 2.
• Three terms (although y 3 term is missing, it is present with coefficient 0).

Visual representation of polynomials

Let's try to visualize the polynomial 2x 2 + 3x + 1.

Explanation: In this SVG representation, each red circle represents the value of the polynomial for a given x-coordinate. The points plotted are (1,6), (2,11) and (3,22), which show how the polynomial grows as x increases.

Types of polynomials

Polynomials can be classified based on the number of terms in them:

• Monomial: A polynomial with a single term. Example: 4x 3
• Binomial: A polynomial with two terms. Example: 3x + 2
• Trinomial: A polynomial with three terms. Example: x 2 + 5x + 6

Operations on polynomials

Adding polynomials

To add polynomials, you combine like terms. Like terms are terms that have the same variable raised to the same power.

Example: Add the polynomials 2x 2 + 3x + 4 and x 2 - 2x + 5.

(2x 2 + 3x + 4) + (x 2 - 2x + 5) = (2x 2 + x 2) + (3x - 2x) + (4 + 5) = 3x 2 + x + 9

Subtracting polynomials

Subtracting polynomials is the same as adding them, except that you subtract the coefficients of like terms.

Example: Subtract the polynomial x 2 + 4x - 2 from 3x 2 - x + 5.

(3x 2 - x + 5) - (x 2 + 4x - 2) = (3x 2 - x 2) + (-x - 4x) + (5 + 2) = 2x 2 - 5x + 7

Multiplication of polynomials

To multiply polynomials, use the distributive property to make sure that each term in the first polynomial multiplies each term in the second polynomial.

Example: Multiply (x + 2) by (x - 3).

(x + 2)(x - 3) = x(x) + x(-3) + 2(x) + 2(-3) = x 2 - 3x + 2x - 6 = x 2 - x - 6

Division of polynomials

Polynomial division can be done using either long division or synthetic division. To stay brief and clear, we will discuss simple long division.

Example: Divide 2x 2 + 3x + 4 by x + 1.

 _________
x + 1 | 2x 2 + 3x + 4
       - (2x 2 + 2x)
          -----------------
             x + 4
            -(x + 1)
          -----------------
                3

The result is 2x + 1 and the remainder is 3.

Remainder theorem and factor theorem

Remainder theorem

The remainder theorem says that if you divide the polynomial f(x) by (x - a), the remainder is f(a).

Example: Find the remainder when dividing f(x) = 2x 3 + 3x 2 - 5x + 4 by x - 2.

f(x) = 2x 3 + 3x 2 - 5x + 4
f(2) = 2(2) 3 + 3(2) 2 - 5(2) + 4 = 16 + 12 - 10 + 4 = 22

Thus, the remainder is 22.

Factor theorem

The factor theorem is an extension of the remainder theorem. It states that (x - a) is a factor of a polynomial f(x) if and only if f(a) = 0.

Example: Check if x - 3 is a factor of x 3 - 6x 2 + 11x - 6.

f(x) = x 3 - 6x 2 + 11x - 6
f(3) = 3 3 - 6(3) 2 + 11(3) - 6 = 27 - 54 + 33 - 6 = 0

Since f(3) = 0, x - 3 is actually a factor.

Graphing polynomials

Graphing polynomials provides a visual representation of the function and its solutions. The degree of the polynomial determines how many possible turns the graph can take.

Graph of a quadratic polynomial

A quadratic polynomial of the form ax 2 + bx + c graphs as a curve called a parabola. The direction of the parabola depends on the sign of the leading coefficient a.

Consider the polynomial f(x) = x 2 - 4x + 3.

This graph is a parabola opening upward. The vertex of the parabola is at the point where the parabola turns.

Graph of cubic polynomial

Cubic polynomials of the form ax 3 + bx 2 + cx + d can have S-shaped graphs. They can have one or two turning points, depending on the coefficients.

Consider the polynomial f(x) = x 3 - 3x 2 - x + 3.

The graph has an S-shape, showing the typical behavior of a cubic polynomial.

Importance of polynomials

Understanding polynomials is not just limited to solving math problems on paper. Polynomials are widely used in various fields such as economics, physics, engineering, computer graphics, and statistics. They are used to model scenarios where entities have changes in rates, provide curves for computer graphics, describe the trajectories of objects in physics, and much more.

Conclusion

Polynomials are an important concept in mathematics whose applications extend far beyond the classroom. Understanding the fundamentals of polynomials, including operations, theorems, and graphing, equips you with the skills to solve and understand more complex mathematical problems. By being able to manipulate and understand polynomials, you unlock a powerful toolset for both academic and real-world problem-solving.

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