Simple Trinomials of the Form x2+bx+c Using Algebra

Tiles and Patterning

Key Points l  Factoring

¡ The process of rewriting an expression as a product or multiplication (i.e.) putting the brackets back.

l  Trinomials ¡ Polynomials with three terms. The first term is called the

quadratic term because it has a coefficient and a variable with an exponent of 2, the second term is called the linear term because it has a coefficient and a variable and the third term is called the constant term because it is a number.

l  Simple Trinomials ¡ Trinomials that have a quadratic term with a coefficient of 1.

Factoring x2+bx+c Using Algebra Tiles l  Represent the expression x2+4x+3 with algebra tiles. (Start

with x2-tile and the 3- unit tiles. Then place the 4-x tiles.)

l  What are the length and width of the rectangle?

Note: There are no other ways to make rectangles with this set of tiles.

The length is x+3 and the width is x+1

Factoring x2+bx+c Using Algebra Tiles (continued) l  The dimensions of the rectangle are (x+1) and (x+3). These

are called the factors. l  The expression x2+4x+3 can be rewritten as (x+1)(x+3). l  You just factored a trinomial! Let’s try another one! l  Example:

1.  Factor x2+5x+6

Is this the only possibility?

Which would result in 5-x tiles?

= (x+2)(x+3)

Factoring x2+bx+c Using Algebra Tiles (continued) l  Examples:

1.  Factor. a)  x2+4x-12 b) x2-7x+10 c) x2-6x-7

= (x-2)(x+6) = (x+1)(x-7) = (x-2)(x-5)

Factoring x2+bx+c Using Patterning l  To factor x2+bx+c follow these steps:

1.  Start the answer with an equal sign. 2.  Write two sets of brackets and inside each

bracket put the variable. 3.  Now examine the trinomial and find the two

numbers (we’ll call them r and s) that are the product of c and whose sum is b.

4.  Write these numbers with their signs in the brackets.

l  Let’s practice finding these two “magic” numbers by completing the chart on the next slide!

r s r + s r x s 4 3 7 12 3 -2 1 -6 6 8 14 48 -4 -2 -6 8 4 5 9 20 5 -2 3 -10 -3 1 -2 -3 -3 -7 -10 21 7 -2 5 -14 6 5 11 30 6 -9 -3 -54 14 -3 11 -42 -6 -9 -15 54

Factoring x2+bx+c Using Patterning (continued) l  Did you notice?

¡  If c is positive and b is positive, then the factors will both be positive. If c is positive and b is negative, then both will be negative. If c is negative, the factors will have opposite signs.

l  You’re ready to factor some trinomials! l  Example:

1.  Factor x2+14x+45

You are looking for two numbers that multiply

together to give 45 and the same two numbers must add

together to give 14. The magic numbers are +9 and +5.

= (x+9)(x+5)

SUM PRODUCT

Factoring x2+bx+c Using Patterning (continued)

l  Examples: 1.  Factor each of the following:

a)  x2-2x-35 b) x2-15x+56 =(x-7)(x+5) =(x-8)(x-7) c)  x2-12x+20 d) x2+3x+2 =(x-10)(x-2) =(x+1)(x+2) e)  x2+3x-28 f) x2-x-30 =(x+7)(x-4) =(x-6)(x+5) g)  x2-4x+4 h) x2+10x+16 =(x-2)(x-2) =(x+8)(x+2) i)  x2+6x-40 j) x2+9x-22 =(x+10)(x-4) =(x+11)(x-2)

Factoring Special Trinomials l  Difference of Squares

¡  In the form, x2 - a2. They have no linear term, it can therefore be written as 0x. In order to get a sum of 0, the factors will have to be the same number with opposite signs.

¡  To find the “magic” numbers, take the square root of the number on the end.

l  Example: 1.  Factor x2-36 Take the square root of 36. It is 6.

The magic numbers are +6 and –6. =(x+6)(x-6)

Factoring Special Trinomials (continued)

l  Examples: 1.  Factor each of the following.

a)  x2-100 b) x2-49 =(x-10)(x+10) =(x-7)(x+7)

c)  x2-16 d) x2-1 =(x-4)(x+4) =(x-1)(x+1)

e)  x2-81 f) x2-121 =(x-9)(x+9) =(x-11)(x+11)

Homework

l  Worksheet