polynomials?

Question

can someone briefly explain how to add, subtract, multiply, and divide polynomials

Answer 1

For addition and subtraction, you perform the operation on like terms. So if you're doing this (3x^2 + 5x -19) - (x^2 -2x +2), then you take care of the x^2's, the x's, etc separately so you do 3x^2 - x^2, and then 5x - - 2x etc and the answer to this problem would be 2x^2 + 7x -21



For multiplication, you must make sure you multiply each term in the first polynomial with each term in the second (like FOIL but maybe with more terms)

e.g. (x^2 + 4) * (2x^2 -5x +6)

(x^2)*(2x^2) + (x^2)*-5x + (x^2)*6 + 4*(2x^2) + 4*-5x + 4*6

combine like terms if you want



For division you can try factoring each polynomial to see if things cancel out. Or you can do it like long division.

e.g. (x^2 - 4)/(x-2)

factor the numerator and it's (x+2)(x-2)/(x-2) and the (x-2)'s cancel and the answer would be x+2

Answer 2

Consider what a number written in base 10 place value system really is.



143 is really:

1x10^2 + 4x10^1 + 3x10^0



let x = 10: and this becomes:

x^2 + 5x + 3 <= look it's a polynomial.



So, you perform all those operations exactly the same as with numbers with one exception: since x is a variable then for certain values the polynomial can be zero hence in that case you cannot use it as a divisor. A little more accurately, you have to note the values of x that make the divisor zero and specifically exclude them.

Answer 3

Addition of Polynomials:

To add polynomials, combine similar terms and proceed to addition of monomials.



Example: (2x + 3y - 6xy) + (-5x + 2y + 3xy)



2x + 3y - 6xy Add the numerical coefficients

using the rule in addition

-5x + 2y + 3xy of signed numbers and bring

down the similar literal

-3x + 5y - 3xy coefficient.



Addition of signed numbers:

The sum of two numbers is a positive number.

The sum of two negative numbers is a negative number.

The sum of a positive and a negative number is the difference

between their absolute values, and the sign of the sum follows the

sign of the number with the greater absolute value.



Subtraction of Polynomials

To subtract polynomials, combine similar terms and follow the steps in subtracting

monomials.



Example: (-2a2 + 5a + 7) - (8a2 + 9a + 14)



-2a2 - 5a + 7 The numerical coefficients

are subtracted as in signed

-8a2 + 9a + 14 numbers, and the literal

coefficient is affixed to the

6a2 + 14a + 21 difference.



Multiplication of a Polynomial by a Monomial

To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial. In other words, apply the distributive property of multiplication.



Example: 3x(2x2 + 5x - 3) = 3x(2x2) + 3x(5x) + 3x(-3)

= 6x3 + 15x2 - 9x



Multiply of a Polynomial by another Polynomial

In multiplying a polynomial by another polynomial, each term in the multiplier is multiplied by each of the terms in the multiplicand. Similar terms in the partial products are combined.



Example: (4x - 5) (3x + 6)



4x - 5

___3x + 6_____

12x2 - 15x → 3x(4x - 5) Add the partial products.

24x - 30 → 6(4x - 5)

12x2 + 9x -30



Division of a Polynomial by a Monomial

To divide a polynomial by a monomial, each term of the polynomial is divided by the monomial. This is the distributive property in division.



Example: -8a2b4c3 + 12ab3c2

2ab2c



-8a2b4c3 + 12ab3c2 = - 8a2b4c3 + 12ab3c2

2ab2c 2ab2c 2ab2c



= -4ab2c2 + 6bc



Division of a Polynomial by a Polynomial

The process of dividing a polynomial by another polynomial is similar to that of dividing whole numbers. It is advisable however, that the polynomials are arranged in descending powers of one of the variables involved.



Example: 7x2 - 10x - 8

x - 2



Step 1 : Divide the first term of the dividend 7x2, by the first term of the divisor x. The

partial quotient is 7x.



Step 2: Multiply 7x by the divisor x-2, obtaining 7x2 - 14x.



Step 3: Subtract 7x2 - 14x from 7x2 -10x. The result is 4x.



Step 4: Bring down -8, obtaining 4x-8 as new dividend.



Step 5: Divide 4x by x. The result is 4.



Step 6: Multiply 4 by x-2.The product is 4x-8 which is subtracted from the dividend 4x-8.

The result is 0.