Multiplying Binomials Mentally

The Distributive Property (x + 5)(2x + 6) (x+5) 2x + (x+5)6 2x(x+5) +6(x+5) 2x² + 10x + 6x + 30 2x² + 16x +30

NOTE : Since there are THREE terms this is called a TRINOMIAL

Trinomials Multiplying MOST binomials results in THREE terms You can learn to multiply binomials in your head by using a method called F O I L

The FOIL Method (x + 4)(x +2) first terms last terms (x + 4) ( x + 2) inner terms outer terms

(x + 4) ( x + 2) Now write the products x² + 2x + 4 x + 8 first outer inner last terms terms terms terms

To Multiply any two binomials and write the result as a TRINOMIAL follow these steps multiply the first two terms multiply the two outer terms multiply the two inner terms multiply the last two terms

Special Cases There are several special cases of multiplying binomials Difference of Squares Perfect Square Trinomials

Difference of two squares When you multiply the sum of two terms and the difference of two terms you get a BINOMIAL (a + b) (a – b) = a² - b² This binomial is the difference of two squares

A Closer Look (k – 4) (k + 4) Using FOIL k ² + 4k - 4k - 16 k ² - 16 (6c – 3) ( 6c + 3) Using FOIL 36c² +18c – 18c – 9

Squaring A Binomial When you square any binomial(that is multiply it by itself) you get a TRINOMIAL (x+9)² means (x+9)(x+9) Using FOIL x² + 9x + 9x + 81 x² + 18x +81 This is called a PERFECT SQUARE TRINOMIAL

REMEMBER: The square of a binomial is the sum of three things: The square of the first term Twice the product of the terms The square of its last term

Perfect Square Trinomials (3x – 6 )² = (3x - 6) (3x – 6) 9x² – 36 x + 36 (2m –4)² = (2m - 4)( 2m – 4) 4m² –16m + 16

An example (6x + 3)² The square of the first term: 36 x² Twice the product of the terms 2( 6x • 3) +36 x The square of the last terms (3)(3) +9 36x² + 36x + 9

Perfect Square Trinomial patterns (a + b)² = a² + 2ab + b² ( a – b)² = a ² - 2ab + b²