Polynomial Long Division and Simplification Techniques

Study Notes

Polynomials and Their Expressions

Polynomials are mathematical expressions that consist of variables raised to different powers and coefficients representing constants. They often appear in algebraic equations and are used to describe various real-world phenomena. In this article, we will explore two key operations related to polynomials: polynomial long division and simplifying polynomial expressions.

Polynomial Long Division

Polynomial long division is similar to dividing integers using long division. It involves breaking down one polynomial into another by repeated subtraction. Here's how it works:

Divide the first coefficient of the dividend by the leading coefficient of the divisor. If the result is less than 0, we take its absolute value. This gives us the quotient (q_0) and the remainder (r_0), which is a polynomial with lower degree than the original divisor.

[ a = q_0d + r_0 ]

Calculate the next coefficient of the quotient, (q_1), by multiplying (q_0) by the leading coefficient of the divisor and subtracting (r_0). This process continues until there is no remainder left, i.e., ({r_{n - 1} = 0}).

The final quotient is given by [q(x) = q_0 + xq_1 + x^2q_2 + ... + x^{n - 1}q_{n - 1}.]

For example, let's perform polynomial long division of (6x^3 - 9x + 7) by (2x - 1):

[ \begin{align*} & 6x^3 - 9x + 7 \ = & (2x - 1)(3x^2 + 3) + (-8x + 7)\ = & (2x - 1)(3x^2 + 3) + (-8x + 7) \ = & (2x - 1)(3x^2 + 3) + (-8x + 7) \ = & (2x - 1)(3x^2 + 3) + (-8x + 7) \ = & (2x - 1)(3x^2 + 3) + (-8x + 7) \ = & (2x - 1)(3x^2 + 3) + (-8x + 7) \ = & 3x^3 + 3x - 8x + 7 \end{align*} ]

So, the quotient is (3x^2 + 3) and the remainder is (-8x + 7).

Simplifying Polynomial Expressions

Simplifying a polynomial expression means finding an equivalent form that includes fewer terms and has simpler coefficients. There are several methods to achieve this:

Combining Like Terms: To combine like terms, you need to have two terms that share the same variable exponent. For example, if you have (5x^2 + 2x^2), you can combine these like terms to get (7x^2).
Distributive Property: The distributive property allows you to rearrange factors within an expression without changing its value. For instance, (5ab = 5a \times b = 5 \cdot ab).
Expanding: Expansion involves multiplying a term containing a sum by each individual factor within the sum. For example, if you have (3(a + 2)), you expand by multiplying each term inside the brackets by 3, giving you (3a + 6).

Here's an example of simplifying a quadratic polynomial expression:

Given three variables, (A, B, C,) we can simplify the expression (5C - A - B) as follows:

[ \begin{align*} & 5C - A - B \ = & 5(C - A/5 - B/5) \ = & 5(C - (A/5 + B/5)) \ = & 5(C - D) \ \end{align*} ] where we defined (D = A/5 + B/5.) This expression is now more concise and easier to work with.

Description

Explore polynomial long division, a method similar to dividing integers, and learn how to simplify polynomial expressions by combining like terms, applying the distributive property, and expanding. Follow along with examples to enhance your understanding of these fundamental operations in algebra.