How are polynomials multiplied? For example, (5x-1)(6x-3)

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caledon eNotes educator| Certified Educator

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`

Wiggin42 | Student

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`

` `

` `

maria-vivanco | Student

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. 

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first number multiplied by other first number

5x-1)(6x-3)             5x*6x

then first multiplied by second

5x-1)(6x-3)                               5x*(-3)

then second muliplied by first

5x-1)(6x-3)                   -1*(6x)

then second multiplied by second

5x-1)(6x-3)                 -1*(-3)

then simplidy and place together


simplify again


crystaltu001 | Student


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

Chantelm | Student

You need to F.O.I.L (first,inner,outer,last) 


5x*6x then -1*6x then 5x*-3 then -1*-3



Wiggin42 | Student


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)

Jyotsana | Student

Box Method``







parama9000 | Student

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.

parama9000 | Student

@maria-vivanco it is negative 3

parama9000 | Student


=6x(5x-1) - 3(5x-1)



Yojana_Thapa | Student

The way I multiply polynomials is by using the box method.` `

`(5x-1)(6x-3) `

`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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fashionableb1 | Student

To multiply a  monomial and a polunomial , use the "Distributive property" and the properties of exponents. Use the Foil method for this one.

maria-vivanco | Student

its actually not -3 its positive three.