Polynomial Tutorial
3. Algebra of Polynomials
By now, you should be familiar with variables and exponents, and you may have dealt with expressions like 3x³ or 6x. Polynomials are sums of these "variables and exponents" expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions.
3.1 Addition
Adding polynomials is just a matter of combining like terms, with some order of operations considerations thrown in. As long as you're careful with the minus signs, and don't confuse addition and multiplication, you should do fine.
Like terms are those terms that contain the same pronumeral or pronumerals.
For example, 8x and 10x are like terms because they contain the same pronumeral in x. Likewise, 8xy and 10xy are like terms because they contain the same pronumerals ( xy).
Clearly, 8xz and 10yz are not like terms because although both terms contain the pronumeral z, the other pronumeral in each term is different. Terms with different pronumerals are called unlike terms.
Ex.
1. 7 ab + 9 ab = 16 ab 2. 14 xy + 6 xy = 20 xy 3. 8 ab + 6 ba = 8ab + 6 ab = 14 ab 4. 2 ab + 9 ab + 7 ba
= 2ab + 9 ab + 7 ab
= 18 ab 5. 6x² + 2x + 4 + 10x² + 5x + 6
= (6x² + 10x²) + ( 2x + 5x) + (4 + 6)
= 16x² + 7x + 10
3.2 Subtraction To subtract Polynomials, first reverse the sign of each term you are subtracting (in other words turn "+" into "-", and "-" into "+"), then add.
Subtracting polynomials is quite similar to adding polynomials. Here are some examples, done both horizontally and vertically:
Simplify (x³ + 3x² + 5x – 4) – (3x³ – 8x² – 5x + 6)
The first thing I have to do is take that negative through the parentheses. Some students find it helpful to put a "1" in front of the parentheses, to help them
By now, you should be familiar with variables and exponents, and you may have dealt with expressions like 3x³ or 6x. Polynomials are sums of these "variables and exponents" expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions.
3.1 Addition
Adding polynomials is just a matter of combining like terms, with some order of operations considerations thrown in. As long as you're careful with the minus signs, and don't confuse addition and multiplication, you should do fine.
Like terms are those terms that contain the same pronumeral or pronumerals.
For example, 8x and 10x are like terms because they contain the same pronumeral in x. Likewise, 8xy and 10xy are like terms because they contain the same pronumerals ( xy).
Clearly, 8xz and 10yz are not like terms because although both terms contain the pronumeral z, the other pronumeral in each term is different. Terms with different pronumerals are called unlike terms.
Ex.
1. 7 ab + 9 ab = 16 ab 2. 14 xy + 6 xy = 20 xy 3. 8 ab + 6 ba = 8ab + 6 ab = 14 ab 4. 2 ab + 9 ab + 7 ba
= 2ab + 9 ab + 7 ab
= 18 ab 5. 6x² + 2x + 4 + 10x² + 5x + 6
= (6x² + 10x²) + ( 2x + 5x) + (4 + 6)
= 16x² + 7x + 10
3.2 Subtraction To subtract Polynomials, first reverse the sign of each term you are subtracting (in other words turn "+" into "-", and "-" into "+"), then add.
Subtracting polynomials is quite similar to adding polynomials. Here are some examples, done both horizontally and vertically:
Simplify (x³ + 3x² + 5x – 4) – (3x³ – 8x² – 5x + 6)
The first thing I have to do is take that negative through the parentheses. Some students find it helpful to put a "1" in front of the parentheses, to help them