Polynomials Part 1

  A monomial is a single term.

    examples: 3x, 5ab, wyz, 2x² , x²y⁶

A polynomial is one or more monomials combined with addition and/or subtraction.

   example:      3x + 5ab, 10x - 6

    We are interested in polynomials with a single independent variable.

examples:

   A binomial has two terms and a trinomial has three terms.

   Polynomials are, by convention written from the highest to the lowest exponents of the independent variable.

example:             2x + 3x² + 4 + 6x³

      is written:     6x³ + 3x² + 2x + 4

      The leading term is the term with the largest exponent, in this case the leading term is 6x³. The leading coefficient is 6. The exponent of the leading term is the degree of the polynomial, and in this case, the degree is 3.

Definition of a polynomial:    (advanced)

Compact notation is used for this using the Greek letter Σ (sigma) for sum like so:

   This notation is read:  the sum of the terms a_ixⁱ  from i = 0 to n.

which is:     a₀x⁰ + a₁x¹ + a₂x² + ... + a_nxⁿ

and, since x⁰ = 1  and we usually do not show the power of 1  we have

for the sum :   a₀ + a₁x + a₂x² + ... + a_nxⁿ

The constant term is a₀.

   Suppose we have the polynomial:

     12 + 2x - 3x² + 2x³

then

     a₀ = 12

     a₁ = 2

     a₂ = -3

     a₃ = 2

The individual coefficients 'a' do not need to be distinct. Notice that a₁ = a₃.

Continuing:

Not all terms need to be present. For example, 5y³ - 2y is missing the square and the constant terms (shown in red next)

5y³ - 2y = 5y³ + 0y² - 2y + 0y⁰

This is an important to note, when dividing polynomials these missing terms  must be included in the dividend polynomial.

Another example:

         -16x⁵ + x - 10

with missing terms becomes

        -16x⁵ + 0x⁴ + 0x³ + 0x² + x - 10

(here we have a₄ = a₃ = a₂ = 0, a₁ = 1 and  a₅ = -16)

Addition/Subtraction of Polynomials

Like terms of polynomials are combined. Like terms have equal exponents (of the independent variable)

Examples:

Example:         add     3x² + 1      to      4x² - x - 5

    like terms are:      3x² and 4x² 

                              1 and -5

So we get    (3x² + 4x²)   - x   +   (1 - 5) = 7x² - x - 4

Example: subtract 2y² - 1 from 4y³ - 3y² + 2

It helps to line up the polynomials by power first.

write missing terms

Example: add:   2x³ + 3x² + x - 10

              to      2x² + x³ - x + 5

(like terms are shown in color)

      2x³ + 3x² + x - 10      +      2x² + x³ - x + 5

Multiplication/Division of Polynomials:

Multiply term by term using the distributive law of multiplication over addition/subtraction.

Example:   multiply   (2x² +x)(x² - x + 5)

Here we have a binomial (2x² +x) multiplying a trinomial (x² - x + 5) .

      The strategy to use is:  take each term in the first polynomial and multiply it by the second polynomial then simplify.  (Here we are distributing the second polynomial over the first.)

    (2x² +x) (x² - x + 5)

     2x² (x² - x + 5)   +    x(x² - x + 5)

Distributing both products we get

  2x²x² - 2x²x + 2x² 5  +  xx² - xx + x5

Simplify

2x²⁺² - 2x²⁺¹ + 10x² + x¹⁺² - x¹⁺¹ + 5x

2x⁴ - 2x³ + 10x² + x³ - x² + 5x

Collecting like terms

2x⁴ + (- 2x³ + x³ ) + (10x² - x²) + 5x

Finally add like terms

Answer: 2x⁴ - x³ +9x² + 5x

Example multiply: ( 3x² + x + 2 )( x² + 3x - 2 )

        Notice that I have used tables to organize the polynomials to carry out addition and subtraction.  As the polynomials get larger tables make it easy to organize the multiplications.

        List the first polynomial down the first column, one term per row in order from highest exponent to the constant term.   The second polynomial is repeated in each row down the second column written in order from highest exponent to the constant term.

       Then carry out the multiplication as shown in the third column and write the product for each in decreasing exponent order lining up the like terms from row to row, then add.

Special Binomial Products

Examples:

1)   (x + y)²  = x² +2xy + y²

2)   (3  +  9)²  =  3² +2 · 3 · 9 + 9²  = 9 +54 + 81 = 144

3)   (2a + 6b)²  = (2a)²  + 2 (2a)(6b) + (6b)² 2

                      =  4a²   + 24ab  + 36b²

4)  (x + y)(x - y) = x²  - y²  

5)  (15 + 1)(15 − 1) = 15² − 1 = 225 − 1 = 224

Practical application:  suppose you wanted to know the product of 17 and 13. Knowing

the square of 15, subtracting 4 from that square gives the answer: 221

17 = 15 + 2,  13 = 15 − 2,  so we have (15 + 2)(15 − 2) = 15²  − 4 = 221

6)   (12c + 3g)(12c − 3g) = (12c)² − (3g)²  = 144c²  − 9g²

7)   (x − y) ²  =  x² − 2xy + y²  

8)   (12 − 9)²   = 12²  − 2(12)(9)  + 9²  = 144 − 216 + 91 = 9

9)   (4h − 3d)²  = (4h)² − 2(4h)(3d) + (3d)²

                      = 16h²  − 24hd + 9d²