How are polynomials multiplied? For example, (5x-1)(6x-3)
Not all polynomials are the same, but when it comes to multiplication, what they have in common is this;
Each number in a bracket must be multiplied by each number in the other brackets.
This means that the first number, 5x, needs to be multiplied by 6x AND by -3 (remember to keep the signs). Likewise, -1 needs to be multiplied by both 6x and -3.
The example is an instance of a two-term polynomial. For these types of polynomial, you can use the FOIL method.
FOIL stands for First, Outer, Inner, Last.
(5x - 1) (6x -3)
First = the first number in each bracket. 5x times 6x.
Outer = the "outside" of the brackets. 5x times -3.
Inner = the "inside" of the brackets. -1 times 6x.
Last = the last number in each bracket. -1 times -3
`5x times 6x = 30x^2`
`5x times -3 = -15x and 6x times -1 = -6x -15x + -6x = -21x`
`-1 times -3 = 3`
combining these results, we get;
`30x^2 - 21x +3`
But lets say you don't have neat binomials. Perhaps you have to multiply a trinomial by a binomial.
You will have to use the 1. Distributive property with each individual term in your binomial and then 2. add all like terms.
`(x^3 + 2x^2 + x) (2x^2 + 3)`
`2x^5 + 4x^4 + 2x^3 `
`3x^3 + 6x^2 + 3x`
`2x^5 + 4x^4 + 5x^3 + 6x^2 + 3x`
Polynomials can be answered in different ways. The way that I have found easier is the box method. So in the picture that i have linked shows what it looks like. So basically you make a box (in your case you would make a 2 by 2 box) You would put the first part of the polynomial on one side and the other on the other side.
5x-1 would go on one side.
6x-3 would go on the other. This method helps you visually see what is being multiplied and and how it all goes together during the FOIL process.
Basically you would multiply everything
5x would multiply with 6x = 30x^2
-1 x 6x= -6x
-3 x 5x = -15x
-1 x -3 = -3
combine like terms and you would get 30x^2 -21x -3
(look at the picture attached its easier to understand)
For any of the polynomials, this would work. This also works for factoring polynomials.
first number multiplied by other first number
then first multiplied by second
then second muliplied by first
then second multiplied by second
then simplidy and place together
You distribute the numbers
5x(6x - 3) + -1(6x - 3)
30x^2 - 15x -6x + 3
Combine like terms
The answer is 30x^2 -21x +3
You need to F.O.I.L (first,inner,outer,last)
5x*6x then -1*6x then 5x*-3 then -1*-3
First Outer Inner Last
5x times 6x = 30x^2 (first)
5x times -3 = -15x (outer)
-1 times 6x = -6x (inner)
-1 times -3 = 3 (last)
30x^2 -15x -6x + 3 (putting it all together)
30x^2 - 21x +3 (Simplify)
From me my suggestion would be to simply split the second or the first bracket, which ever you prefer, so that do can get a clearer picture of what you need to multiply, and then do the multiplication.
@maria-vivanco it is negative 3
=6x(5x-1) - 3(5x-1)
The way I multiply polynomials is by using the box method.` `
`5x * 6x= 36x^2`
`5x * -3 = -15x`
`-1 * 6x = -6x `
`-1 * -3 = 3 `
`36x^2-15x-6x+3 = 36x^2 -21x+3`
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To multiply a monomial and a polunomial , use the "Distributive property" and the properties of exponents. Use the Foil method for this one.
its actually not -3 its positive three.