POLYNOMIALS return

The following are example of Polynomials.

4x^{3 } + 3x^{2} - 5x + 4

3t^{2} - 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:

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

3t^{2}(2t^{3}
+ 4t - 3) -2t(2t^{3} + 4t - 3) +7(2t^{3} + 4t - 3) =

6t^{2+3}
+ 12t^{2+1} - 9t^{2} - 4t^{3+1 } - 8t^{1+1}
+ 6t + 14t^{3}
+ 28t - 21 =

6t^{5}
+ 12t^{3} - 9t^{2} - 4t^{4 } - 8t^{2} + 6t + 14t^{3}
+ 28t - 21

The next step is to simplify by combining like terms

6t^{5}
+ 12t^{3} - 9t^{2} - 4t^{4 } - 8t^{2} + 6t + 14t^{3}
+ 28t - 21 =

6t^{5}
+ 12t^{3} + 14t^{3} - 4t^{4 } - 8t^{2} -9t^{2}
+ 6t
+ 28t - 21 =

6t^{5}
+ 26t^{3} - 4t^{4 } - 17t^{2} +34t - 21

Next we want to arrange the terms in descending order:

6t^{5}
+ 26t^{3} - 4t^{4 } - 17t^{2} +34t - 21 = 6t^{5}
- 4t^{4 }+ 26t^{3} - 17t^{2} +34t - 21

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

For example try these problems (r^{7 } +
3r)(r^{4} - r^{3}) = r^{7} (r^{4}
- r^{3}) + 3r(r^{4} - r^{3}) =
answer

3t^{2}( t^{3} - 4t^{2} + 2t - 4) =
answer

ADDITION OF POLYNOMIALS

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

2r^{4} + 3r^{3} - r^{2} + 5r + 4
+ 3r^{4} + 4r^{2} -2r - 7

2r^{4} + 3r^{4} + 3r^{3} + ( -r^{2}
) +4r^{2} + 5r + (-2r) + 4 + (-7) = 5r^{4} + 3r^{3}
+ 3r^{2 } +3r + (-3) = 5r^{4} + 3r^{3} + 3r^{2 } +3r
- 3

Try these: (4t^{3} + 2t^{2} - 7t + 5) +
(5t^{4} + 3t^{3} + 4t - 7) answer

(5p^{4} - 3p^{2} + 2p) + (2p^{3 } - 4p^{2}
+ 4p - 6) answer

SUBTRACTION OF POYNOMIALS

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

(2r^{4} + 3r^{3} - r^{2} + 5r + 4)
- (3r^{4} + 4r^{2} -2r - 7) = (2r^{4} + 3r^{3} - r^{2} + 5r + 4)
+ (-3r^{4} - 4r^{2} + 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 2r^{4} + 3r^{3} - r^{2} + 5r + 4
+ (-3r^{4} ) + (- 4r^{2} ) + 2r + 7) = 2r^{4}
+ (-3r^{4} ) + 3r^{3} - r^{2} + (- 4r^{2} ) + 5r
+ 2r + 7 + 4

-r^{4} + 3r^{3} -
5r^{2} + 7r + 11

Try these: (4t^{3} + 2t^{2} - 7t + 5) -
(5t^{4} + 3t^{3} + 4t - 7)
answer

(5p^{4} - 3p^{2} + 2p) - (2p^{3 } - 4p^{2}
+ 4p - 6) answer

DIVISION OF POLYNOMIALS

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

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

(x^{2 } - 4x - 21) ÷ (x + 3)
(x^{2 } - 4x - 21) is called the dividend and (x + 3) is called the
divisor.

The first term of the dividend, x^{2} , is divided by
the first term of the divisor, x. x^{2} ÷ x = x

The divisor, (x + 3) is multiplied by x and the result
subtracted from the dividend. x(x + 3) = x^{2} + 3x

(x^{2 } - 4x - 21) - (x^{2 } + 3x)
= x^{2 } - 4x - 21 - x^{2 } - 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 (x^{2 } -
4x - 21) ÷ (x + 3) is x - 7

Try these problems (x^{2
} + 13x + 36) ÷ ( x + 9) answer

(4x^{3} + 5x^{2} + 2x + 4) ÷ (x + 2)
answer