POLYNOMIALS return

 The following are example of Polynomials.

                        4x³  + 3x² - 5x + 4

                        3t² - 2t + 7 trinomial

                        4r - 6  binomial

                                 3m  monomial

All of the above are polynomials with the last 3 having specific names based on the number of terms they contain. Trinomials have 3 terms, binomials have 2 terms and monomials have 1 term.

MULTIPLICATION OF POLYNOMIALS:   

(3t² - 2t + 7)(2t³ + 4t - 3) is a mathematical expression telling us to multiply each term in the second set of parenthesis by each term in the first.            

                                        3t²(2t³ + 4t - 3) -2t(2t³ + 4t - 3) +7(2t³ + 4t - 3) =

                                                      6t²⁺³ + 12t²⁺¹ - 9t² - 4t³⁺¹  - 8t¹⁺¹ + 6t + 14t³ + 28t - 21 =

                                                      6t⁵ + 12t³ - 9t² - 4t⁴  - 8t² + 6t + 14t³ + 28t - 21

                                                             The next step is to simplify by combining like terms

                                                      6t⁵ + 12t³ - 9t² - 4t⁴  - 8t² + 6t + 14t³ + 28t - 21  = 

                                                      6t⁵ + 12t³ + 14t³ - 4t⁴  - 8t² -9t² + 6t  + 28t - 21 =

                                                      6t⁵ + 26t³ - 4t⁴  - 17t² +34t - 21

                                                 Next we want to arrange the terms in descending order:

                                                      6t⁵ + 26t³ - 4t⁴  - 17t² +34t - 21 =  6t⁵ - 4t⁴ + 26t³  - 17t² +34t - 21

The above procedure is used for any polynomial including binomials and monomials.

For example try these problems  (r⁷  + 3r)(r⁴ - r³)   =   r⁷ (r⁴ - r³) + 3r(r⁴ - r³) =    answer

                                                       3t²( t³ - 4t² + 2t - 4) = answer

ADDITION OF POLYNOMIALS

(2r⁴ + 3r³ - r² + 5r + 4)  +  (3r⁴ + 4r² -2r - 7)  (REMEMBER WE CAN ONLY ADD LIKE TERMS - 2r⁴ and 3r⁴ are like terms because the variables and exponent are the same. 3r³  and 4r²  are unlike terms because their exponents are different and hence cannot be combined. )

2r⁴ + 3r³ - r² + 5r + 4  +  3r⁴ + 4r² -2r - 7

2r⁴ + 3r⁴ + 3r³ + ( -r² ) +4r² + 5r +  (-2r)  + 4 + (-7) = 5r⁴ + 3r³ + 3r²  +3r + (-3) = 5r⁴ + 3r³ + 3r²  +3r - 3

Try these:  (4t³ + 2t² - 7t + 5) + (5t⁴ + 3t³ + 4t - 7)  answer

                  (5p⁴ - 3p² + 2p) + (2p³  - 4p² + 4p - 6)  answer 

SUBTRACTION OF POYNOMIALS

LET USE THE ABOVE PROBLEM BUT CHANGE IT TO A SUBTRACTION PROBLEM:

(2r⁴ + 3r³ - r² + 5r + 4)  -  (3r⁴ + 4r² -2r - 7)  = (2r⁴ + 3r³ - r² + 5r + 4)  +  (-3r⁴ - 4r² + 2r + 7)  In algebra we change

subtraction problems to addition problems by changing the - to a + and then change the sign of every term that

immediately follows the minus sign.  Then proceed as if it is an addition problem by combining like terms                                                               2r⁴ + 3r³ - r² + 5r + 4  +  (-3r⁴ ) + (- 4r² ) + 2r + 7) =  2r⁴ + (-3r⁴ ) + 3r³ - r² + (- 4r² ) + 5r  + 2r + 7 + 4                                                         

-r⁴ + 3r³ - 5r² + 7r + 11

Try these:  (4t³ + 2t² - 7t + 5) - (5t⁴ + 3t³ + 4t - 7)  answer

                    (5p⁴ - 3p² + 2p) - (2p³  - 4p² + 4p - 6)   answer

DIVISION OF POLYNOMIALS

4t⁵ + 6t³ + 8 ÷ 2t²  = 4t⁵ ∕ 2t² + 6t³ ∕2t²  + 8 ∕ 2t²  =  2t^{5 - 2} + 3t^{3 - 2} + 4 ∕ t²  = 2t³ + 3t  + 4 ∕ 2t²  In this case the divisor, 2t² , is one term and hence each term in 4t⁵ + 6t³ + 8 can be divided individually. 

When the divisor is two or more terms the division is a little more difficult to do.  For example:

    (x²  - 4x - 21) ÷ (x + 3)    (x²  - 4x - 21) is called the dividend and (x + 3) is called the divisor.

The first term of the dividend, x² , is divided by the first term of the divisor, x.    x² ÷ x = x

The divisor, (x + 3) is multiplied by x and the result subtracted from the dividend. x(x + 3) = x² + 3x

(x²  - 4x - 21) - (x²  + 3x)  = x²  - 4x - 21 - x²  - 3x = -7x - 21    Next the first term of the answer, -7x,

 is divided by the first term of the divisor, x.     -7x ÷ x = -7   The answer -7 is multiplied by the divisor

(x + 3).  -7(x + 3) = -7x - 21  This is subtracted from the answer from the first division  (-7x - 21)

Therefore (-7x - 21) - (-7x - 21) = -7x - 21 + (7x + 21)  = -7x - 21 + 7x + 21 = 0 

So the answer for the division problem  (x²  - 4x - 21) ÷ (x + 3) is x - 7

Try these problems   (x²  + 13x + 36) ÷ ( x + 9)   answer

                                   (4x³ + 5x² + 2x + 4) ÷  (x + 2)   answer   

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