﻿ POLYNOMIALS

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

(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  =  2t5 - 2 + 3t3 - 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

return