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

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caledon's profile pic

caledon | High School Teacher | (Level 3) Senior Educator

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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`

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Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

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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`

` `

` `

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maria-vivanco | Student, Grade 11 | (Level 1) Valedictorian

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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)(6x-3)

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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acompanioninthetardis's profile pic

acompanioninthetardis | Student, Undergraduate | (Level 1) Valedictorian

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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

30x^2-15X-6X+3

simplify again

30X^2-21x+3

crystaltu001's profile pic

crystaltu001 | Student, Grade 10 | (Level 1) Valedictorian

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

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

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Chantelm | Student, Grade 10 | (Level 1) Salutatorian

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

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

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

30x^2-6x-15x+3

30x^2-21x+3

Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

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(5x-1)(6x-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)

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Jyotsana | Student, Grade 10 | (Level 1) Valedictorian

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Box Method``

5x*6x=30x^2

5x*-3=-15x

-1*6x=-6x

-1*-3=3

30x^2-15x-6x+3

30x^2-21x+3

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parama9000 | Student, Grade 11 | (Level 1) Valedictorian

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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.

Yojana_Thapa's profile pic

Yojana_Thapa | Student, Grade 10 | (Level 1) Valedictorian

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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, Grade 10 | (Level 1) Salutatorian

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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.

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