Objectives:  To recall how to add, subtract, multiply and factor   polynomials

Recall 1Definition of polynomial.

An expression made from the  sum, difference or product of numbers and integer powers  of an unknown variable x is called a polynomial in x.   .
Polynomials are usually written in a descending order of the powers of x; however, it  is not necessary to write a polynomial in such an order. The following  are examples of polynomials

3*x^2 + 5*x + 2,

x^5 + x^4 + x^3 + x^2 + x + 1,

5*x^4 - 3*x^2 - 7*x + 1.

Note that any constant may be interpreted as a polynomial; for example,  5 is the polynomial 5*x^0.

Recall 2: A polynomial does not necessarily have to be formed from a  single variable x.  Polynomials in a involve powers of a.  For example,

5*a^4 - 3*a^2 - 7*a + 1

is a polynomial in a.  Polynomials may involve several variables; for  example,

5*x^4 - 3*y^2 - 7*z + 1,

and

3*x^3*y2 - 4*y^3*z - 7*x*z^3

are polynomials in x, y and z.

Recall 3: The non-variable numbers are called the coefficients of  the polynomial.  For example, the 5, the -3, the -7 and the 1 are the  coefficients of the polynomial

5*x^4 - 3*y^2 - 7*z + 1.

Coefficients of polynomials need not be integers or rational numbers.   For example, the following is also considered to be a polynomial.

(5/2)*x^4 - 3*y^2 -sqrt(7)*z +pi

Recall 4: The terms of a polynomial are separated by addition signs.   For example, in the polynomial above, the terms are

(5/2)*x^4,  - 3*y^2,  -sqrt(7)*z and pi

The term  pi is really the term pi*x^0.  In general the term involving x^0 is  called the constant term.

Important note

If a variable has no coefficient in front of it, then the coefficient is understood to be the number 1.   For example, all the coefficients of the polynomial
x^5 + x^4 + x^3 + x^2 + x + 1

are 1.  In other words the polynomial could have been written as

1*x^5 + 1*x^4 + 1*x^3 + 1*x^2 + 1*x + 1.

Negative signs in a polynomial are attached to the coefficients.  For example, in the polynomial

(5/2)*x^4 - 3*y^2 -sqrt(7)*z +pi ,

the coefficient of the second and third terms are -3 and -sqrt(7), respectively.  In other words, the polynomial should be thought of as a sum of terms, some of which are positive and some of which are negative.

More terminology

The degree of a term is the sum of the exponents of that term.  For example,  the degree of the first term of the polynomial

3*x^3*y^2 - 4*y^3*z - 7*x*z^3

is 3 + 2 or 5.

The degree of the polynomial is the largest degree of any of the terms of the polynomial.
The polynomial

(5/2)*x^4 - 3*y^2 -sqrt(7)*z +pi

is of degree 4, while the polynomial

3*x^3*y^2 - 4*y^3*z - 7*x*z^3

is of degree 5.

Like terms of a polynomial are terms that contain the same variable raised to the same powers.  For example, the first and third terms of the polynomial

3*x*y^2 - 4*y^3*z - 5*y^2*x + 1

are like terms.

Recall 5: To add two polynomials add their like terms.  For example,

(3*x^2 + 5*x + 2) + (x^5 + x^4 + x^3 + x^2 + x + 1)

equals

x^5 + x^4 + x^3 + 4*x^2 + 6*x + 3.

Recall 6: Multiplication of two polynomials is a little more  complicated, but not difficult.  First, every term of the first polynomial  must be multiplied by every term of the second polynomial.  For  example, to multiply 3*x + 5 by 2*x - 3, write out the products of all  combinations of the terms.  They are

(3*x)*(2*x) = 6*x^2                 5*(2x) = 10*x

(3*x)*(-3) = -9*x                     5*(-3) = -15

Next, add these terms together to get

6*x^2+ 10*x + (-9)*x -15.

Finally, it is good form to simplify the expression by combining the like  terms (namely the two middle terms).  We get:

6*x^2+ 1*x -15.

For more complicated polynomials, it is best to multiply by columns.    Multiplication by columns works just like multiplication of real  numbers.  The exponents order the columns.  To multiply

3*x^2+ 5*x - 6   by   2*x^2 - 3*x + 4

we write one polynomial under the other, multiply each term of the top  polynomial by each term of the bottom polynomial, organizing terms in  columns of like terms, then adding the columns of like terms.  The  organized multiplication takes the form illustrated below.

3*x^2 + 5*x -  6
2*x^3 - 3*x + 4
6*x^4 + 10*x^3 - 12*x^2
-9*x^3 - 15*x^2 + 18*x
12*x^2 + 20*x - 24
6*x^4 +  1*x^3 -  15*x^2 + 38*x - 24

Examples

Example 1.      Add (  6*x^4 + 10*x^3 - 12*x^2 ) to (-9*x^3 - 15*x^2 + 18*x).

Solution:        The second polynomial does not have an x^4 term and the first polynomial does not have an x term.   However, we  may write the two polynomials as

6*x^4 + 10*x^3 - 12*x^2+  0*x
and
0*x^4 -  9*x^3 -  15*x^2 + 18*x.

Now we may add these column by column, to get

6*x^4 +  1*x^3 -  27*x^2 + 18*x.

Notice that the exponents act as place holders, ordering the   columns.

Example 2.      Determine the degree of the polynomial

5*x^7*y^3*z^4 - 3*y^3*x^5*z - 15*x*z^3.

Solution:        The three terms have degrees 14, 9 and 4,   respectively.  The polynomial is of degree 14, since 14 is the  largest degree.

6*x^4 + 10*x^3 - 12*x^2
-9*x^3 - 15*x^2 + 18*x
and  12*x^2 + 20*x - 24.

Solution:       Line up the three polynomials so that each column  contains only like terms.

6*x^4 + 10*x^3 - 12*x^2
-9*x^3 - 15*x^2 + 18*x
12*x^2 + 20*x - 24.

Then add the columns to get

6*x^4 +  1*x^3 - 15*x^2 + 38*x - 24.

Example 4.   Multiply 3*x^2 + 5*x + 2 by 2*x^2 - 3*x - 4.

Solution: Write out the products of all the terms.  They are

3*x^2*2*x^2 =  6*x^4     5*x*2*x^2 = 10*x^3     2*2*x^2 = 4*x^2

3*x^2*(-3*x) = -9*x^3      5*x*(-3x) = -15*x^2      2*(-3*x) = -6*x

3*x^2*(-4) = -12*x^2        5*x*(-4) = -20*x            2*(-4) = -8

Adding the results of all these multiplications we get the polynomial

6*x^6 + x^3 - 23*x^2 - 26*x - 8.

Note

There are several special products of polynomials that you should be familiar with.  They will appear often in future exercises and in future applications.  The next examples are about three of those special products.

Example 5.  Multiply (x + 5) by (x - 5).

Solution:   x  + 5
x  -  5
x^2 + 5*x
- 5*x - 25
x^2 + 0*x - 25

Notice that the result can be written as x^2 - 25.   The thing to notice here is that we started with two very similar polynomials; they were of the form

x + a
and         x - a.

The result was a polynomial of the form x^2 - a^2.  It had a term of degree 2 and a constant term, but no term of degree 1.  This is a very special product of two polynomials, one in which the degree 1 term disappears.   In general,

(x + a)*(x - a) = x^2 - a^2.

Example 6 Multiply (x + 3) by (x + 3).

Solution:    x  + 3
x  + 3
x^2 + 3*x
+ 3*x + 9
x^2 + 6*x + 9

The thing to notice here is that in general,

(x + a)*(x + a) = x^2 + 2*a*x + a^2.

Example 7  Multiply (x - 1) by (x^2 + x + 1).

Solution: x^2 + x + 1
x - 1
x^3 + x^2 + x
- x^2 -  x - 1
x^3+0*x^2 - 0*x - 1

Notice that the result can be written as x^3 - 1.

Note  In general,

(x - a)*(x^2 + a*x + a^2) = x^3 - a^3.

The next two examples show how to apply this to specific cases.

Example 8  Multiply (x - 2) by (x^2 + 2*x + 4).

Solution:   x^2 + 2*x + 4
x -  2
x^3 + 2*x^2 + 4*x
- 2*x^2 -  4*x - 8
x^3+ 0*x^2 -   0*x - 8

Notice that the result can be written as x^3 - 8.

Example 9  Multiply (x + 2) by (x^2 - 2*x + 4).

Solution: If we view this as the product

[x - (-2)]*[x^2 + (-2)x + (-2)^2],

then we may use the equality (x - a)*(x^2 + a*x + a^2) = x^3 - a^3 for a = -2 to guess that this product is x^3 - (-2)^3 = x^3 +8.
We check on this by making the calculation

x^2 - 2*x + 4
x - (-2)
x^3 - 2*x^2 + 4*x
2*x^2 -4*x +8
x^3+ 0*x^2 -   0*x - 8

Notice that the result can be written as x^3 - 8.

EXERCISES

Perform the indicated operation:

1. (5*x^2 + 3*x^3 - 3) + (3*x^3 - 5*x^2 + 4*x + 2)

2. (2*x^2 + 5*x^3 - 8*x) - (3 x^3 - 4*x^2 + 3*x + 6)

3  (2*a^3 + 6*a^2 + 8*a + 5) - (5*a^3 - 7*a^2 + 5*a + 6)

4 (2*x^2*y^3 + 5*x^3*z + 8*x*y) - (3 x^3 - 4*x^2*y^3 + 3*x*y + 6)

5 (3*x^99 + 99*y^3) + (99*x^3 + 3*y^99)

6.  (2*x^2 + 5)  (3*x^2 - 6)

7.  (4*x^2 - 6*y^2)  (5*x^2 - 3*y^2)

8.  (x^2 - 6*x + 6) (x^2 + 3*x + 3)

9.  (x - 12) (x + 12)

10.  (x - 9) (x^2 + 9*x + 81)

11.  (2*a - 3)  (4*a^2 + 6*a + 9)

12.  (x^2 + 3*y) (x^4 - 3*x^2*y + 9*y^2)

13.  (5*x^3 + 3*x^2 - 3) + (3*x^3 - 5*x^2 + 4*x + 2) - (2*x^3 + 3*x - 3)

14.  4*x^2 *(x^6 + 3*y^4) (x^6 - 3*y^4)

15.  (x - 12)  (x + 12)  (x + 3) .

16.  (x - 4)  (x^2 + 4*x + 16) - (x^3 - 64)