MULTIPLYING A BINOMIAL BY A TRINOMIAL

To multiply a binomial by a trinomial, you can use the Distributive Property several times. 

Multiply (5x + 3) by (2x2 + 10x - 6).

You can also use a rectangle model to multiply a binomial by a trinomial. This is similar to finding the area of a rectangle with length (2x2 + 10x - 6) and width (5x + 3).

Product of monomials are written in each row and column. 

To find the product, add all of the terms inside the rectangle by combining like terms and simplifying, if necessary.   

10x3 + 6x2 + 50x2 + 30x - 30x - 18

=  10x3 + 56x2 - 18

Another method that can be used to multiply a binomial by a trinomial is the vertical method. This is similar to methods used to multiply whole numbers. 

Example 1 :

Multiply.

(x + 3)(x2 - 5x + 7)

Answer :

=  (x + 3)(x2 - 5x + 7)

Distributive. 

=  x(x2 - 5x + 7) + 3(x2 - 5x + 7)

Distribute again. 

=  x(x2) + x(-5x) + x(7) + 3(x2) + 3(-5x) + 3(7)

Simplify. 

=  x3 - 5x2 + 7x + 3x2 - 15x + 21

Combine the like terms.

=  x3 - 5x2  + 3x2+ 7x - 15x + 21

=  x3 - 2x2 - 8x + 21

Example 2 :

Multiply.

(a + b)(2a2 - 5ab + 3b2)

Answer :

=  (a + b)(2a2 - 5ab + 3b2)

Distributive. 

=  a(2a2 - 5ab + 3b2) + b(2a2 - 5ab + 3b2)

Distribute again. 

=  a(2a2) + a(-5ab) + a(3b2) + b(2a2) + b(-5ab) + b(3b2)

Simplify. 

=  2a3 - 5a2b + 3ab2 + 2ab - 5ab2 + 3b3

Combine the like terms.

 =  2a3 - 5a2b + 2ab - 5ab2 + 3ab2+ 3b3

 =  2a3 - 3a2b - 2ab2 + 3b3

Example 3 :

Multiply.

(2x + 3y)(x2 - xy + y2)

Answer :

=  (2x + 3y)(x2 - xy + y2)

Distribute. 

=  2x(x2 - xy + y2) + 3y(x2 - xy + y2)

Distribute again.

=  2x(x2) +2x(-xy) + 2x(y2) + 3y(x2) + 3y(-xy) + 3y(y2)

Simplify. 

=  2x3 - 2x2y + 2xy2 + 3x2y - 3xy2 + 3y3

Combine the like terms.

 =  2x3 + x2y - xy2 + 3y3

Example 4 :

Multiply.

(m - n)(m2 + mn + n2)

Answer :

=  (m - n)(m2 + mn + n2)

Distribute.

=  m(m2 + mn + n2) - n(m2 + mn + n2)

Distribute again. 

=  m(m2) + m(mn) + m(n2) - n(m2) - n(mn) - n(n2)

=  m3 + m2n + mn2 - m2n - mn2 - n3

Combine the like terms.

=  m3 - n3

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