Polynomials

The following is a polynomial in x:

a_{n}x^{n}+a_{\left ( n-1 \right )}x^{\left ( n-1 \right )}+...+a_{1}x+a_{0}

The polynomial is of degree n*, a_{n} is the leading coefficient, and a_{0} is the constant term.

* the exponent must be a positive integer.

Important Vocabulary:

1.  polynomial – an algebraic expression a number of terms

2.  coefficient – the number in front of the variable

3.  degree – the power of a variable

4.  monomial – a polynomial of only one term

5.  binomial – a polynomial of two terms

6.  trinomial – a polynomial of three terms

7.  standard form – a polynomial written with descending powers of x

Examples:                                 Standard Form                    Degree               LC*

a)\, \, 4x^{2}-5x^{7}-2+3x                 -5x^{7}+4x^{2}+3x-2                  7                      -5

b)\, \, 4-9x^{2}                                                            -9x^{2}+4                                                           2                                      -9

c)\, \, 8                                                                            8\left ( 8=8x^{0} \right )                                                      0                                        8

*LC – leading coefficient

Degree of a polynominal with more than 1 variable:

Degree of a term = sum of the exponents of the variables in the term

Degree of a polynomial = highest degree of its terms

Leading coefficient = coefficient of the highest-degree term

More Examples:

1.  a polynomial of degree zero — 12

2.  a trinomial of degree five — -3x^{5}+2x^{3}+x

3.  a binomial with a leading coefficient of -2 — 1-2x^{3}

4.  a monomial of with a positive degree — 3x^{2}

5.  a trinomial with a leading coefficient of 2/3 –  \frac{2}{3}x^{4}+x^{2}+10

6.  a third degree polynomial with a leading coefficient of 1 — x^{3}+3x^{2}+3x+1

 

Operations with Polynomials

Addition/Subtraction – sum/difference – put like terms together

Multiplication – products

  • Polynomial by polynomial

                \left ( 3x-2 \right )\left ( 5x+7 \right )

                3x\left ( 5x+7 \right )-2\left ( 5x+7 \right )

                15x^{2}+21x -10x-14

                15x^{2}+11x-14

  • Special Products

Sum & Difference of same terms          \left ( a+b \right )\left ( a-b \right )=a^{2}-b^{2}

Square of a binomial                               \left ( a+b \right )^{2}=a^{2}+2ab+b^{2}  

                                                                  \left ( a-b \right )^{2}=a^{2}-2ab+b^{2}

Cube of a binomial                                 \left ( a+b \right )^{3}=\left ( a+b \right )\left ( a+b \right )\left ( a+b \right )

                                                                              =\left ( a+b \right )\left ( a^{2}+2ab+b^{2} \right )

                                                                              =a^{3}+2a^{2}b+ab^{2}+a^{2}b+a^{2}b+2ab^{2}+b^{3}

                                                                              =a^{3}+3a^{2}b+3ab^{2}+b^{3}

                                              and             \left ( a-b \right )^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}

Factoring — writing a polynomial as the product of factors

For information about factoring go to the post called Factoring.

Filed under: Pre-requisites for Pre-Calculus

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