NORTH CAMPUS

Prepared by:  Carlos I. Gil,  Department of Mathematics,  Miami Dade College, North Campus

Miami, February 27, 2005

 

Polynomials

 

Consider the expressions:  

   a)    (polynomial of first degree or linear polynomial)  

b)     (polynomial of second degree or quadratic polynomial)

c)      (polynomial of third degree or cubic polynomial)

   d)   ( polynomial of fourth degree or quartic polynomial or bi-quadratic)

   e)   (polynomial of fifth degree or quintic polynomial)

    f)   (polynomial of sixth degree or sextic polynomial or hexic)

    g)          (polynomial of seventh degree or septic polynomial or heptic)

    h)     (polynomial of eighth degree or octic polynomial)

    i)     (polynomial of ninth degree or nonic polynomial)

    j)    (polynomial of tenth degree or decic polynomial)

    k)    (polynomial of one hundredth degree or hectic polynomial)

 

They are all examples of polynomials in the variable x.  Notice that every polynomial consists of a sum of terms.  A term consists of numbers and variables in juxtaposition (one next to the other), which indicates multiplication.  The variables contain exponents, which must be positive integers.  The sum of the exponents on the variables is called the degree of the term. 

Examples of terms:  (a)   is a term of degree 6;   (b)   is a term of degree 7;   (c)   is a term of degree 12;

The expressions ,  ,  ,    are not terms.  Why?

 

Polynomials are considered simple algebraic expressions in the sense that they include only the operations of addition, subtraction, and multiplication.  The exponents of the variable x are all positive integers.  The degree of a polynomial is the largest degree of all the terms included in the polynomial.

 

The general expression that represents a polynomial of degree n in the variable x is ,  where n is a non-negative integer and an ¹ 0.  The term  is called dominant term (or leading term).  The real numbers  are called coefficients of the polynomial.  The real number  is called the leading coefficient and the real number  is called the constant term.  The non-zero constant c, (a non-zero real number), can be considered a polynomial of degree zero and is referred to a the constant monomial.  The number 0 is not considered a polynomial (0 is not considered a constant monomial).  

 

It is important to realize that a polynomial could include more than one variable.  Examine the following examples:

1)      x2 + y2  is a polynomial of degree 2 in the variables x and y.

2)      x3 y2 z + x2 y3 z2 – 2xyz3 is a polynomial of degree 7 in the variables x, y, and z.

 

CLASSIFICATION OF POLYNOMIALS BASED ON THEIR DEGREES

Linear:  A polynomial of degree one is said to be a linear polynomial.

Examples:  (A)  3 – 7x                                               (B) 

 

Quadratic:  A polynomial of degree two is said to be a quadratic polynomial.

 Examples:  (A)  1 – 3x – 4x2       (B)               (C)   x2 + 4x             (D)   9x2 – 25

 

Cubic:  A polynomial of degree three is said to be a cubic polynomial.

 Examples:  (A)  4x3 – 2x2 + x + 5     (B)  x3        (C)   x2 – 5x3      (D)  x2 + 8     (E)  2x3 – x – 1

 

Quartic:  A polynomial of degree four is said to be a quartic polynomial (or biquadratic polynomial)

 Examples:  (A)  3x4 – 5x3 + x2 – 2x + 6           (B)   -2x4            (C)   x – 3x2 – 5x3 + 2x4         (D)  x4 + 81

                     (E)  x4 – 13x2 + 36                         (F)   9x4 – x2

                    

Quintic:  A polynomial of degree five is said to be a quintic polynomial.

 Examples:  (A)  9x5 – 3x4 + x3 – x2 + 7x – 3    (B)   x5                (C)   2x – x3 + x5        (D)  x5 + 32

                     (E)  8x5 – 3x4 + 6x3 – 7x2              (F)   9x4 – x2       (G)  

 

Sextic:  A polynomial of degree six is said to be a sextic polynomial, (or hexic polynomial)

 Examples:  (A)    2x6 + 4x5 – 3x4 + 5x3 – 6x2 – 3x + 1            (B)   9x6

                     (C)   1 – x + x2 – x3 + x4 - x5 + x6                        (D)   x6 + 26x3 – 27

                     (E)    0.75x6 – 1.50x4 + 2.33x2 – 3.56                   (F)    64 – x6

 

Septic:  A polynomial of degree seven is said to be a septic polynomial, (or heptic polynomial)

 Examples:  (A)    4x7 + 3x6 – x5 – 2x4 + 5x3 – 7x2 – 9x + 10         (B)   – 8x7

                     (C)   1 – x + x2 – x3 + x4 - x5 + x6 – x7                       (D)   x7 + x6 – 6x5

                     (E)    1.5x7 – 2.5x5 + 4.3x3 – 3.6x + 5.1                      (F)    25x7 – 9x5

 

Octic:  A polynomial of degree eight is said to be an octic polynomial.

 Examples:  (A)    -x8 + 6x7 + 4x6 – 4x5 – 3x4 + 8x3 – 9x2 – 10x + 1      (B)   – 10x8

                     (C)   1 – x + x2 – x3 + x4 - x5 + x6 – x7 + x8                        (D)   x8 + 3x7 – 10x6

                     (E)    0.5x8 – 3.4x6 + 5.3x4 – 6.6x2 + 2.1                              (F)    16 – 9x8

 

Nonic:  A polynomial of degree nine is said to be a nonic polynomial.

Examples:  (A)  -3x9 + 2x8 – x7 + 1.33x6 – 51x5 + 4x4 + 103x3 – 8x2 + 7x + 60       (B) 

                    (C)  25x + 3x3 – x4 + 2x5 – 9x8 + 10x9                                                    (D)  x9 – 27

 

Decic:  A polynomial of degree ten is said to be a decic polynomial.

Examples:  (A)  6x10 -2x9 + 5x8 – x7 + 3.33x6 – 5x5 + 42x4 + 3x3 – 18x2 + x + 110         (B) 

                    (C)  5x + 13x3 – 25x4 + 112x5 – x8 + 7x9 – 100x10                                         (D)  x10 – 1

 

Hectic:  A polynomial of degree one hundred is said to be a hectic polynomial.

Examples:  (A)  3x100 -2x99 + 55x98 – x77 + 1.55x63 – 45x51 + 420x46 + 13x33 – 18x29 + x17 + x2 + 2

                    (B)  41x100                                  (C)  4x + 13x31 – 256x48 + 12x59 – x82 + 77x94 – 10x100

                    (D)  x100 – 100                            (E)  2x100 – x50 – 1

 

CLASSIFICATION OF POLYNOMIALS BASED ON THEIR NUMBER OF TERMS

Monomial:  A polynomial of one term is said to be a monomial.

Examples:  (A)           (B)              (C)  –3xy2              (D) 

 

Binomial:  A polynomial of two terms is said to be a binomial.

 Examples:  (A)  1 – 3x                (B)  x4 – 27x             (C)   x2 + 4x             (D)   9x3 – 25x

 

Trinomial:  A polynomial of three terms is said to be a trinomial.

 Examples:  (A)  – 2x2 + x + 5         (B)  x12 – 7x6 – 8         (C)   x4 – 5x3 + 3x2         (D)  2x3 – x – 1

 

Polynomial of n terms:  A polynomial of four or more terms is said to be a polynomial of n terms, where n is a “place holder” for the cardinal number of terms.

 Examples:    (A)   2x4 – 3x3 – 5x2 + 2x  is said to be a polynomial of 4 terms  (or a 4-term polynomial).

(B)    3x4 – 5x3 + x2 – 2x + 6  is said to be a polynomial of 5 terms  (or a 5-term polynomial).

(C)      is said to be a polynomial of 10 terms (or a 10-term polynomial).

… and so on…

 

OPERATIONS WITH POLYNOMIALS

Addition and Subtraction:  Combine like terms.  Examples are shown below.

1)      Add   (4x2 – x + 3) + (5x2 – 2x – 4)

Solution:  (4x2 – x + 3) + (5x2 – 2x – 4)  =  (4x2 + 5x2) + (-x – 2x) + (3 – 4)  =  9x2 – 3x – 1

 

2)      Subtract  (4x2 – x + 3) - (5x2 – 2x – 4)

Solution:  First transform into a sum and then add

                     (4x2 – x + 3) - (5x2 – 2x – 4)  = (4x2 – x + 3) + (-5x2 + 2x + 4)  =  -x2 + x + 7

 

3)      Simplify completely:  2(3x – 2x2 + x3) – 4(1 – x2) – (5x3 – 2x2 + 4x – 3)

Solution:  Use the order of operations

                2(3x – 2x2 + x3) – 4(1 – x2) – (5x3 – 2x2 + 4x – 3)  =

                (6x – 4x2 + 2x3) + (-4 + 4x2) + (-5x3 + 10x2 – 4x + 3)  =  -3x3 + 10x2 + 2x – 1

 

 Multiplication:  Use rules of exponents and the distributive property of real numbers.

(Monomial)´(Monomial):      1)  

                                                2)  

                                                3)  

                                                4)  

 

(Monomial)´(Binomial):       1)  

                                                2)  

                                                3)  

 

(Monomial)´(Trinomial):      1)  

                                                2)  

                                                3)  

 

(Monomial)´(Polynomial):    1)  

                                                2)  

                                                3)  

                                                       =  

 

(Binomial)´(Binomial):        1)  

                                                2)  

                                                3)  

                                                4)    (x + y)2  =  (x + y)(x + y)  =  x2 + xy + xy + y2   =   x2 + 2xy + y2   

                                                5)    (x + y)(x – y)  =  x2 – xy + xy – y2  =  x2 – y2   (Difference of Squares)

 

(Binomial)´(Trinomial):      

1)  

2)  

3)      (Sum of Cubes)

4)      (Difference of Cubes)

 

(Trinomial)´(Trinomial):  

1)  

2)   

 

Other Products:

1)     =  x3 – 2x2 – 4x + 8

2)     =  x6 – 6x5 + 3x4 + 44x3 – 105x2 + 90x – 27

3)     = 

 

The  WolframResearch company is a distributor of the computer program WebMathematica and allows you to try some explorations of this program on line by visiting their WebSite.  Click on WebMathematica to link to their site and try long multiplications of polynomials.  For example try    and click on DO IT to obtain .  The resulting polynomial in factored form is .

 

SPECIAL PRODUCTS

1)  Distributive Property:  a(x1 + x2 + x3 + … + xn)  =  ax1 + ax2 + ax3 + … + axn.  Notice that a becomes a common factor, (the greatest common divisor or gcd of the resulting expression.

      Examples:  1.1)   3x(2xy – 5y2 + 30)  =  3x(2xy) – 3x(5y2) + 3x(30)  =  3x2y – 15xy2 + 90x

                                 Notice that 3x is the greatest common divisor (gcd) of the resulting expression.

                      

                        1.2)   -7a3b ( 8a2 – 6b2 + 5ab )  =  –7a3b( 8a2) – 7a3b(–6b2) – 7a3b(5ab)

                                                                           =  -56a5b + 42a3b3 – 35a4b2

                                  Notice that  -7a3b  is the gcd of the resulting expression.

 

2)  Product of Conjugates:  (a + b)(a – b)  =  a2 – b2.  Two binomials are said to be conjugates if they contain exactly the same terms with opposite signs between them.  That is, the conjugate of (a + b) is (a – b) and the conjugate of  (a – b)  is (a + b).  The product of such conjugates is a2 – b2 and it is called difference of squares.

      Examples:  2.1)    (x + 3)(x – 3)  =  (x)2 – (3)2  =  x2 – 9

                       2.2)    (4y – 5)(4y + 5)  =  (4y)2 – (5)2  =  16y2 – 25

                       2.3)    (7a + 2b)(7a – 2b)  =  (7a)2 – (2b)2  =  49a2 – 4b2

                       2.4)    (x2 – 6)(x2 + 6)  =  (x2)2 – (6)2  =  x4 – 36

 

3)  Square of a Binomial:  (a + b)2 = a2 + 2ab + b2.  The trinomial  a2 + 2ab + b2  is called perfect squared trinomial.

     Similarly,  (a – b)2 = a2 - 2ab + b2.  The trinomial  a2 - 2ab + b2  is also a  perfect squared trinomial.

     Examples:  3.1)   (x + 5)2  =  (x)2 + 2(x)(5) + (5)2  =  x2 + 10x + 25

                      3.2)    (x – 5)2  =  (x)2 – 2(x)(5) + (5)2  =  x2 – 10x + 25

                      3.3)    (3x + 4)2  =  (3x)2 + 2(3x)(4) + (4)2  =  9x2 + 24x + 16

                      3.4)    (3x – 4)2  =  (3x)2 - 2(3x)(4) + (4)2  =  9x2 - 24x + 16

                      3.5)    (2a + 3b)2  =  (2a)2 + 2(2a)(3b) + (3b)2  =  4a2 + 12ab + 9b2

                      3.6)    (2a - 3b)2  =  (2a)2 - 2(2a)(3b) + (3b)2  =  4a2 - 12ab + 9b2

                      3.7)    (x2 + 6)2  =  (x2)2 + 2(x2)(6) + (6)2  =  x4 + 12x2 + 36

                      3.8)    (x2 - 6)2  =  (x2)2 - 2(x2)(6) + (6)2  =  x4 - 12x2 + 36

 

4)  Cube of a Binomial:  (a + b)3 = a3 + 3a2b + 3ab2 + b3.  The polynomial  a3 + 3a2b + 3ab2 + b3 is called perfect cubed polynomial.

     Similarly,  (a - b)3 = a3 - 3a2b + 3ab2 - b3.  The polynomial  a3 - 3a2b + 3ab2 - b3 is also a  perfect cubed polynomial.

     Examples:  4.1)   (x + 4)3  =  (x)3 + 3(x)2(4) + 3(x)(4)2 + (4)3 =  x3 + 12x2 + 48x + 64

                      4.2)    (x – 4)3  =  (x)3 - 3(x)2(4) + 3(x)(4)2 - (4)3 =  x3 - 12x2 + 48x - 64

                      4.3)    (2a + 5)3  =  (2a)3 + 3(2a)2(5) + 3(2a)(5)2 + (5)3  =  8a3 + 60a2 + 150a + 125

                      4.4)    (2a – 5)3  =  (2a)3 - 3(2a)2(5) + 3(2a)(5)2 - (5)3  =  8a3 - 60a2 + 150a - 125

                      4.5)    (3x + 4y)3  =  (3x)3 + 3(3x)2(4y) + 3(3x)(4y)2 + (4y)3  =  27x3 + 108x2y + 144xy2 + 64y3

                      4.6)    (3x - 4y)3  =  (3x)3 - 3(3x)2(4y) + 3(3x)(4y)2 - (4y)3  =  27x3 - 108x2y + 144xy2 - 64y3

                      4.7)    (x2 + 2)3  =  (x2)3 + 3(x2)2(2) + 3(x2)(2)2 + (2)3 =  x6 + 6x4 + 12x2 + 8

                      4.8)    (x2 - 2)3  =  (x2)3 - 3(x2)2(2) + 3(x2)(2)2 - (2)3 =  x6 - 6x4 + 12x2 - 8

 

5)   Other Products of Binomials:  (x + a)(x + b) = x2 + (a + b)x + ab.  The resulting trinomial x2 + (a + b)x + ab, is called a quadratic trinomial with leading coefficient 1.

      More generally, (ax + b)(cx + d) = acx2 + (ad + bc)x + bd.  The resulting trinomial acx2 + (ad + bc)x + bd, is called a general quadratic trinomial.

                      5.1)   (x + 3)(x + 5)  =  x2 + (3 + 5)x + (3)(5)  =  x2 + 8x + 15

                      5.2)   (x - 2)(x - 4)  =  x2 + (-2  +  -4)x + (-2)(-4)  =  x2 - 6x + 8

                      5.3)   (x - 1)(x + 3)  =  x2 + (-1  +  3)x + (-1)(3)  =  x2 + 2x - 3

                      5.4)   (5x + 2)(3x - 4)  =  15x2 - 14x - 8

                      5.5)   (4x - 1)(2x - 5)  =   8x2 - 22x + 5

 

6)   Other Products:  (a + b)(a2 – ab + b2)  =  a3 + b3.  The resulting binomial a3 + b3 is called the sum of cubes.

       Similarly,   (a - b)(a2 + ab + b2)  =   a3 – b3 .  The resulting binomial a3 – b3 is called the difference of cubes.

       Also:  (x – 1)(xn + xn - 1 + xn - 2 + … + 1)  =  xn + 1  – 1 .

       Examples:  6.1)    (x + 4)(x2 – 4x + 16)  =  x3 + 64

                        6.2)    (x – 2)(x2 + 2x + 4)  =  x3 – 8

                        6.3)    (x – 1)(x4 + x3 + x2 + x + 1)  =  x5 – 1

 

       More generally,    =  xn + 1  -  an + 1

       Example:  (x – 3)(x3 + 3x2 + 9x + 27)  =  x4 – 81

 

Knowledge and correct use of the above special products will help us save time when performing computations involving them and other expressions later.

 

EXERCISES

1)   Circle the letters in front of the expressions that are not polynomials.

      (A)   x                                                                    (B)   1 + 2x – x2

      (C)                                                                 (D)  

      (E)   2x-2 + x + 3                                                   (F)   5x0.2 + 5x2 – 2

      (G)                                                             (H)  

      ( I )                                                 (J)  

      (K)                                                        (L)  

 

2)   Consider the polynomial   

(A)    Give the degree of this polynomial

(B)     How many terms does this polynomial have?

(C)    List the coefficients of this polynomial

(D)    Give the value of the leading coefficient

(E)     Give the value of the constant term

(F)     Write the dominant term

 

3)   Complete the following chart by classifying the given polynomials as requested

      

Polynomial

Name by Degree

Name by Number of terms

2x – 3x2

 

 

8x5 – 3x4 + 4x3 + 8x2 – 5x – 81

 

 

 

 

9x3 + 6x2 – 5x + 1

 

 

625x4 + 0.5x2 – 8

 

 

7x – 5

 

 

 

4)   Which of the following are monomials?   Circle all that apply.

       (A)   2x3y + 1          (B)   x2 + y2 + z2          (C)             (D)             (E)  

5)   Find the degree, the coefficient, and list the variables of   

 

6)   Find the degree of the polynomial   49x2y – 14xy4z + 42x3y2

 

7)   Add or subtract as indicated.

7.1)    (2x2 – 6x + 3)  +  (8x2 – 7x + 5)                 7.2)    (2x2 – 6x + 3)  -  (8x2 – 7x + 5)

7.3)    5(6 – 3x + 5x2) – 3(4x2 – 3x – 7)               7.4)    4(5x3 – 2x2 + x – 3) – 6(3x + 1) – 7(x3 – 2x2 + 8x)

7.5)                            7.6)   

7.7)    (-x4 + 2x3y – 3xy2) – (x4 + 2x2y – xy2)       7.8)   (0.51x2 + 1.61x – 0.73) + (6.41x2 – 2.38x – 0.57)

 

8)   Perform the indicated multiplications and simplify.

       8.1)    3x2y (6x5y)                                                 8.2)   

       8.3)    4x2 (2x3 + 5x)                                             8.4)    5ab (7a3 – 8b2 + 6a2b3)

       8.5)    -6x3y (3x2y – 2xy3 + xy – 4)                      8.6)   

       8.7)    (4x + 7)(x – 3)                                            8.8)    (5x – 3y)(2x + 5y)

       8.9)    (2x3 + 3xy)(4x2 – xy3)                                8.10)   (x + 5)(3x2 – 2x + 4)

      8.11)   (3x + 2y)(x2 – 4xy + y2)                              8.12)   (2x – 3)(4x2 + 6x + 9)

      8.13)   (x4y – 3xy3)(x5y + 2x3y3 – 3xy5)                 8.14)   (5a6 – 6b6)(3a4b – 4ab4 – a3b3)

      8.15)   (2x2 – 3x + 4)(x2 – 2x – 5)                         8.16)   (a3 + 2a2b2 – b3)(a2 – a3b3 + b2)

      8.17)   (a2 – 2a + 1)(a3 – 3a2 + 3a – 1)                  8.18)   (4x3 – 2x2 + 3x – 1)(2x4 – x3 + 2x2 – 3x)

      8.19)   (x + h + 1)2                                                 8.20)   (3x + 3h – 2)2

 

9)   In each case, use the appropriate special product to perform the indicated operations.

       9.1)    (x + 1)2                                                       9.2)    (a – 2)2

       9.3)    (x + h)2                                                       9.4)    (a - h)2

       9.5)    (3x – 2)2                                                     9.6)    (4x – 3y)2

       9.7)    (x + 3)3                                                       9.8)    (x + h)3

       9.9)    (2x – 5)3                                                     9.10)   (3x + 2y)3

      9.11)   (x + 2)(x – 2)                                              9.12)   (3x – 1)(3x + 1)

      9.13)   (4x + 5y)(4x – 5y)                                      9.14)   (6 + y)(6 – y)

      9.15)   (x + 3)(x2 – 3x + 9)                                    9.16)   (2x + 5)(4x2 – 10x + 25)

      9.17)   (x – 6)(x2 + 6x + 36)                                  9.18)   (3x – 4)(9x2 + 12x + 16)

      9.19)   (x – 1)(x3 + x2 + x + 1)                               9.20)   (x – 1)(x7 + x6 + x5 + x4 + x3 + x2 + x + 1)

      9.21)   (x – 2)(x4 + 2x3 + 4x2 + 8x + 16)               9.22)   (x – 4)(x5 + 4x4 + 16x3 + 64x2 + 256x + 1024)

      9.23)   (2x – y)(2x + y) – (x – y)2                           9.24)   (3a + b)2 – (a – 2b)(a2 + 2ab + 4b2)

      9.25)   (a – b)3 – 3ab2(2a – a2b + b2)                     9.26)   (3a + b)(9a2 – 3ab + b2) – (a + b)3

      9.27)   (3x – 4y)(2x + y) – (x – 2y)2                       9.28)   (x – h)2 – 2xh(x2 – 3xh – 4h2) 

 

 

 

Prepared by:  Carlos I. Gil

Miami, February 27, 2005