Manipulation of Simple Polynomials

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Contents

Learning Objectives

After learning this chapter, students will understand:

  1. the concept of positive integral indices, monomials, polynomials and the terminology involved
  2. how to do addition, subtraction and multiplication of polynomials

Two Laws of Positive Integral Indices

For any real numbers `a` and positive integers `m` and `n`, the following laws hold:

  1. `a^m*a^n=a^(m+n)`
  2. `a^m/a^n=a^(m-n)`, where `a!=0`

Monomials

A monomial is an algebric expression with only one term, and which can be written in one of the following forms:

  1. a number, e.g. `6`, `-4`, `0.5` and `7.1`,
  2. a variable with positive integral index, e.g. `x`, `y^3` and `z^7`, or
  3. the product of a number and variable(s), each with a positive integral index, e.g. `6x`, `-4y^3` and `0.5xy^2z^3`.

For a monomial containing variable(s), its numerical part is called the coefficient of the monomial.
For instance:

  1. The coefficient of `x^2` in `4x^2` is `4`,
  2. The coefficient of `xy` in `-2xy` is `-2`.

The sum of the exponents(or indices) of all the variables in a monomial is called the degree of the monomial.
If the degree is `n`, the monomial is called the monomial of degree `n`.
For instance:

  1. The degree of `4x^2` is `2`, i.e. `4x^2` is a monomial of degree 2.
  2. The degree of `-4x^2y^3` is equal to `2+3` = `5`, i.e. `-4x^2y^3` is a monomial of degree 5.
  3. The degree of `7.1` is 0, i.e. `7.1` is a monomial of degree 0.

Polynomials

A polynomial is either a monomial or the sum of two or more monomials. For instance:

  1. `7.1`
  2. `-4x+2`
  3. `a^2+2ab+b^2`

Each monomial in a polynomial is called a term of the polynomial, and the term that does not contain any variable is called the constant term.
The highest degree of all the term(s) in a polynomial is called the degree of the polynomial.
For instance, for the polynomial `a^5-2 a^2 b+b^2-9`,

  1. the coefficient of `a^2 b` is `-2`,
  2. the constant term is `-9`,
  3. the degree of the polynomial is `5`,
  4. the number of terms is `4`.

Like terms and Unlike terms

The terms that contain the same variable(s) to the same power(s) are called like terms. Otherwise, they are called unlike terms.
Polynomials containing like terms can be simplified by adding or subtracting the coefficients of the like terms.
For instance,

  1. `x+2y+3x-5y = (x+3x)+ (2y-5y) = 4x-3y`,
  2. `2a+3a b^2-4a^2 b-3a-a^2 b-2a b^2 = 2a-3a + 3a b^2 -2a b^2-4a^2 b-a^2 b = -a+a b^2-6a^2 b`.

Arrangement of terms

A polynomial containing only one variable is called a polynomial in one variable, and the terms are usually arranged in descending powers(from large to small) or ascending powers(from small to large) of the variable.
For instance,

  1. arrange the terms of the polynomial `2x+3x^3-4-6x^4` is ascending powers of `x`: `-6x^4+3x^3+2x-4`,
  2. arrange the terms of the polynomial `2x+3x^3-4-6x^4` is descending powers of `x`: `-4+2x+3x^3-6x^4`.

Value of a polynomial

When the values of all the variables in a polynomial are given, we can find the value of the polynomial simply by substitution(s).
For instance,

  1. Given that `a=-1`, `b=2`, the value of `-a+a b^2-6a^2 b = -(-1)+(-1)(2)^2-6(-1)^2 (2)= 1-4-12 = -15`.

Addition and subtraction of polynomials

Addition and subtraction of polynomials can be performed by combining like terms.
For instance,

  1. `(4y^2-2y-1)-(-3-2y+2y^2-3y^3)=4y^2-2y-1+3+2y-2y^2+3y^3=3y^3+2y^2+2`.

Multiplication of polynomials

We can multiply polynomials by applying the distributive law of multiplication: `a(x+y)=ax+ay`.
For instance,

  1. `(2x+3y)(x-y)=(2x+3y)x+(2x+3y)(-y)=2x^2+3xy-2xy-3y^2=2x^2+xy-3y^2`.
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