Operations With Polynomials and Radicals

Published: Aug 13th, 2020

Learning Objectives:
By the end of this section, you should be able to:
1. Simplify and evaluate expressions
2. Multiply, divide, add, and subtract radicals

In this lesson, we will sharpen our algebra skills and take a closer look at polynomials and radicals. Because this section is more computational than theoretical, it is especially important to consult the practice problems to consolidate your learning.

Polynomials:

A polynomial is a type of mathematical expression that combines multiple terms separated by addition, subtraction, multiplication, and/or division. Something awesome about math, unlike most other subjects, is that very little memorization is required. For example, the definition of a polynomial can be derived from its name:  "poly", meaning many, and "nomial", meaning terms. A binomial, for example, has two terms; a trinomial has three terms, and so on.

Let's start with a simple example:

Simplify \(4x^2 + 3x + 7x + x^2\).

We do this by grouping the like terms; here, these are the terms that share same degree in the exponent. Not sure how to identify the degree of an exponent? Simply assume that a variable written on its own is implied to have the exponent 1.

We can therefore rewrite the expression as \(4x^2. +3x^1 + 7x^1 + x^2\).

Now, group the terms that share the same degree (number above the variable) of the exponent. This gives: \(5x^2 + 10x^1\), or simply \(5x^2 + 10x \). Done!

Now, let's add in a multiplication term: 

Simplify the expression \((5x)(3x) + 4xy + (x)(y) + x^2\).

Remembering the order of operations (brackets; exponents; division, multiplication, addition, subtraction), we begin with multiplication. Multiplying exponents means to add their degrees (the superscript numbers above their variables).

We can rewrite the expression as \((5x^{1})(3x^{1}) + 4x^{1}y^{1} + (x^{1})(y^{1}) + x^2\).

Multiplying, we get: \(((3)(5)x^{1+1}) + 4x^{1}y^{1} + (xy) + x^2\). We can simplify this by removing the brackets, doing the addition in the exponents, and multiplying the constants. This gives: \(15x^2 + 4xy + xy + x^2\).

Now, we group the like terms as we did before. Here, we have two types of terms: \(x^2\) and \(xy\). Grouping the like terms, we get: \(16x^2 + 5xy\).

Let's add in a divison term.

Simplify the expression \(\frac{5xy + x^3y}{x} +x^2y + 3y\).

A good way to deal with such terms is to see if anything can be factored out; that is, something that is common between the numerator (above the division symbol) and the denominator (below the division symbol), called a common factor. Let's see in this case: 

\(\frac{(x)(y)(5 + x^2)}{x} +x^2y + 3y\). Aha! we can factor out an \(x\) and \(y\) from the numerator. You might notice that the \(x\) term is common between the numerator and denominator. Let's common factor it (remove it from the numerator and denominator). We can do this because this variable represents some number, such as 5. As we know, any number divided by itself gives 1, so dividing \(x\) by itself gives 1. Multiplying any number by 1 gives the number itself, so by doing this, we are restating the same expression in a simpler way:

\(5y + x^2y +x^2y + 3y\). Adding the common terms together, we get \(8y + 2x^2y\). I don't know about you, but I'd take that over \(\frac{5xy + x^3y}{x} +x^2y + 3y\) any day.

Radicals:

A radical is the positive square root of a number. We need to add in the "positive square root" part because taking the square root of a (positive) number will always give two numbers, the positive and negative root. Examples of radicals include \(\sqrt{25}\)\(\sqrt{65}\)\(\sqrt{69}\) (8 something, right?).

Anyway, radicals are quite fun to simplify, as long as you know follow a few rules:

  • \(\sqrt{a} + \sqrt{a} = 2\sqrt{a}\) Adding two of the same radical gives two times that radical. This is just like \(a + a = 2a\)
  • \(2\sqrt{a} - \sqrt{a} = \sqrt{a}\) Subtraction is as we would expect.
  • \(\sqrt{a} \times \sqrt{a} = \sqrt{a^2} = a\). This is one way to simplify a radical. However, the numbers inside the radical are often not the same. In this case: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b } = \sqrt{ab}\).
  • \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).

When simplfiying radicals, be on the look out for perfect squares: 4, 9, 16, 25, 36, ...; numbers whose square root is a natural number (see the lesson "Types of Numbers" for a review on the different... types of numbers!). For example, we can rewrite \(\sqrt{48}\) as \(\sqrt{16 \times 3}\), or \(\sqrt{16} \times \sqrt{3}\), or \(4\sqrt{3}\). The same applies to \(\sqrt{125}\), which can be written as \(\sqrt{25 \times 5}\), or \(5\sqrt{5}\). A radical with a coefficient is called a mixed radical.

That's it! Because this is mostly a computational lesson, there are several practice flashcards included with this lesson. Be sure to try a few for yourselves! 

Comments

    Leave a Reply