A polynomial is an algebraic expression which is obtained when we express the sum of any non-similar monomials.

If we think back to the section on adding monomials we know that they cannot be added when they are not similar. In this case what we get is a polynomial.

Example 8.- The following expressions are all polynomials:

a) 4ax⁴y³ + x²y + 3ab²y³

b) 4x⁴ -2x³ + 3x² - 2x + 5

In the first case the polynomial consists of the sum of three monomials, each one being one of the terms of the polynomial. There are three terms and each one includes more than one letter. In the second case the polynomial has 5 terms. If a term simply consists of one number it is called an independent term (5 in this case but there isn't one in example a).

When a polynomial consists of two monomials it is called a binomial: x²y + 3ab²y³ ; 2x + 3 are two binomials

When it consists of three monomials it is called a trinomial: 8a) and -2x³ + 3x² + 5 are both trinomials.

When there are more than three terms (monomials) it is generally called a polynomial.

With respect to the degree of a polynomial, it takes the highest degree of the monomials in the expression.

Therefore in 8a) the degrees of the monomials (the sum of the exponents of the letters) are 8, 3 and 6 so the degree of the polynomial is 8.

In 8b) the degree is 4.

The numbers which act as factors of the letters (coefficients of the monomials) are also the coefficients of the polynomial: 4 , -2 , 3 , -2  and 5 respectively in 8b).

The sum of polynomials is similar to that of monomials seen earlier in this unit. Similar terms (monomials) can be added together in the polynomials.

"From now on we shall just be working with polynomials with one single letter (x) as these are the most common type"

Example 9.- In order to find the sum of the polynomials:

(4x⁴ - 2x³ + 3x² - 2x + 5 ) + ( 5x³ - x² + 2x )

We just need to add the corresponding terms of degrees 3, 2 and 1 in both polynomials and leave the rest of the terms as they are.

We can also show the sum as follows:

4x⁴ - 2x³ + 3x² - 2x + 5
+ --- 5x³ --- x² +2x
_____________________
4x⁴ + 3x³ + 2x² + -----5

Therefore: In order to add two or more polynomials we need to add corresponding terms together.

If we want to subtract instead of add we could simply change the sign of all the terms in the second polynomial and add the two expressions together.

Example 10.- In order to find the difference or subtract the two polynomials from our earlier example:

(4x⁴ - 2x³ + 3x² - 2x + 5 ) - ( 5x³ - x² + 2x )

We find the sum: (4x⁴ - 2x³ + 3x² - 2x + 5 ) + ( - 5x³ + x² - 2x ) = 4x⁴ - 7x³ + 4x² - 4x + 5

The following window shows the addition and subtraction of two polynomials of the third degree maximum, where it is possible to change the coefficients of each of them. Remember that if one of the coefficients is 0 then the corresponding term has a value of 0, so we do not add or subtract. If there is a term "missing" then we can take the coefficient as being 0.

a) ( - x³ + 5x² - x + 1 ) + ( 5x² - x - 3 )

b) ( 6x² - x + 4 ) + ( 5x³ - x - 1)
The addition of 6a) is shown in the window. Change the coefficients (c1 to c4 for the first polynomial and c5 to c8 for the second) in the window to find the subtraction of 6a) and both the addition and subtraction of 6b).

In order to find the product of two polynomials we need to multiply all the monomials of one expression by all the monomials of the other and add the results. ("Be extra careful when finding the product of powers with the same base number")

If one of the two polynomials is a monomial then the operation is straight forward, as shown in the following window, where the coefficients can be changed.

If both polynomials consist of various terms then we can set the multiplication out like a 'long multiplication' of a big number, taking care to write similar terms underneath each other and leave spaces for 'missing' terms.

The following box illustrates an example of the product of two polynomials with various terms.

Example 11.-

In practice the multiplication is not usually set out as it is in the box, but all the terms are written out in full and then like terms are added together. For example: 

Example 12.- ( - 2x³ + 3x² - 2x + 5 ) · (x + 1) = (-2x⁴ +3x³ -2x² + 5x - 2x³ + 3x² - 2x + 5) = - 2x⁴ + x³+ x² +3x + 5.

Some operations with polynomials are of special interest as they often appear in problems.

the most common are:

Squaring a binomial: addition (a + b)² or subtraction (a - b)²

Obviously, squaring is the same as multiplying the binomial by itself, so:

(a + b)² = (a + b ) · (a + b) = a² + ab + ba + b² = a² + 2ab + b²

"Squaring a sum is the same as the square of the first term plus two times the first term times the second term plus the square of the second term".

Or: (a + b)² = a² - 2ab + b² (as before but changing the sign in the middle).

"We should always remember that the first term "a" could be negative and therefore the middle sign will change". "In general we can treat it like a sum and give each term the sign which precedes it (see example 13b) )".

Example 13.-

a) (2x + 3y)² = (2x)² + 2 · 2x · 3y + (3y)² = 4x² +12xy + 9y²

b) (- x + 3)² = (-x)² + 2 · (-x) · 3 + 3² = x² - 6x + 9

The sum multiplied by the difference: refers to the product of the sum of two monomials multiplied by the difference between them:

(a + b) · (a - b) = a² - ab + ba + b² = a² - b²

We should always remember that "the sum multiplied by the difference is equal to the difference of the squares".

Other special cases which are less common are:

Finding the cube of a sum: (a + b)³ = a³ + 3a²b + 3ab² +b³

Squaring a trinomial: (a + b + c)² = a²+ b² +c² + 2ab+ 2ac + 2bc

	Leoncio Santos Cuervo
	Spanish Ministry of Education. Year 2001