Multiplying Polynomials?

Question

I am having a hard time understanding how to do this...can someone help me please im trying to figure out how to use the distributive property while multiplying polynomials...can anyone explain it to me please....
Thanks.

Answer 1

The easiest way to remember this is to use the FOIL (First, Outside, Inside, Last) method for multiplying two binomials first, then practice with larger polynomials.

Say we are given (X+1)(X+7)
Multiplying the outside numbers, we have X*X =X^2
Multiplying the outside numbers, we have X*7 =7X
Multiplying the inside numbers, we have 1*X =X
Multiplying the last numbers, we have 1*7 = 7
Adding those together, we get X^2 +7X+X+7, or X^2+8X+7.

A couple others to practice with:

(X-7)(X+4)

(3X+1)(X+2)

(X-5)(X-9)

When you multiply longer polynomials, just take one term at a time, and multiply it out.

(X+2)(X^2-3x+4)

So take the first X term and multiply that by each term in the second polynomial
X*X^2=X^3
X*(-3X)=-3X^2
X*4=4X
Then take the second term of the binomial, and multiply that by each term in the second polynomial.
2*X^2=2X^2
2*(-3X)=-6X
2*4=8

Add all of these together and combine like terms:
X^3+(-3X^2)+4X+2X^2+(-6X)+8
X^3-X^2-2X+8

Practice for these kinds of problems is the key. The more you do these kinds of problems, the easier they will become. Good luck!

Answer 2

It would be easier to show you step by step with an example, so here it goes:

Lets start with (x^2 +3x -5) * (4x^2 -2x -7)

First, take the first term in the first polynomial (x^2) and multiply it by all the separate terms in the second polynomial (4x^2, -2x, -7). The result is 4x^4 -2x^3 - 7x^2. This is the first part of your answer.

Next, multiply the second term from the first polynomial (3x) by the terms in the second polynomial. The result is 12x^3 - 6x^2 -21x. This is the second part of the answer.

Then, multiply the last term of the first polynomial (-5) by the term of the second polynomial. The result is -20x^2 +10x + 35. This is the last part of the answer.

Now, combine the parts of the answer to get 4x^4 - 2x^3 -7x^2 +12x^3 -6x^2 - 21x -20x^2 +10x + 35.

Combine like terms and you get your final answer of 4x^4 + 10x^3 - 33x^2 - 11x + 35.

Answer 3

I wish you had included some examples. It would make it easier to explain. I'll make up a few.

3(x-5)=3x-15

3x(x-5)=3x^2-15x

(x-4)(x-6)

first multiply the first x by the (x-6) then multiply the -4 by the (x-6)

x^2-6x -4x +24

x^2-10x +24

Multiply the first term in the first set of parentheses by teh whole second set of parentheses, then multiply the next term in the first set by the whole second set, and if there is a third term, then multiply it by the whole second set of parentheses.

Hope I've helped

Answer 4

Multiply each term in one polynomial by each term in the other polynomial. You will often develop like terms which must be combined.

(x + a)(x + b)

You can line them up like a regular multiplication

x + a
x + b
------
x^2 + ax
. . . + bx + ab
-----------------
x^2 + (a + b)x + ab

You can use two rules:

1) a(b + c) = ab + ac

2) ab = ba

(x + a)(x + b)

z = (x + a) {make a simple substitution}

z(x + b)
zx + zb {rule 1}
(x + a)x + (x + a)b {undo substitution}
x(x + a) + b(x + a) {rule 2}
x^2 + ax + bx + ab {rule 1}
x^2 + (a + b)x + ab {collect like terms}

x^2 + 3x - 5
. . . . 2x + 7
--------------
2x^3 + 6x^2 - 10x
. . . . + 7x^2 + 21x - 35
----------------------------
2x^3 + 13x^2 + 11x - 35


Each term in one by each term in the other.

FOIL only works for binomials. HOMES is more useful, I think. (the great lakes)

Answer 5

(9x + 3)(4x +2)
FOIL
mulitiply the First terms: 9x * 4x = 36x^2
multiply the Second terms: 9x * 2 = 18x
mulitply the Inside terms: 3 * 4x = 12x
multiply the Last terms: 3 * 2 = 6
the resulting term is: 36x^2 + 18x + 12x +6
add like terms: 36x^2 +30x + 6

Answer 6

I'll give an example for binomials. Everything else follows from this just with more terms

(x+y)(2x+3y)
x(2x+3y)+y(2x+3y)
2x^2+3xy+2xy+3y^2
2x^2+5xy+3y^2