Complex Trinomial

Complex Trinomial.

Another of factoring polynomials is called "Simple Trinomials". (ax^2 + bx + c)

example; 2x^2 + 9x + 4 Complex Trinomial

x^2+5x+4 Simple Trinomial

trinomial which starts with a coefficient more than 1. ( where the a is more than 1)

In the following example 2x^2 + 6x + 4

Always common factor if possible.

Factor out 2

2(x^2 +3x+2)

Find factors of 2 that add up to 3.

2(x+ 2) (x+1)

To check your answer:

So, now if you check my the first value in the first bracket and last bracket in second the product of 1x. Then, when you multiply the second value from the first bracket and multiply it by the first value in the second bracket it gives you the product of 2x. When you add the two numbers, 1x + 2x = 3x which is also you your "b" value in the example.

When you are trying to factor trinomials of this form: ax^2 + bx +c

You have to find 2 numbers that multiply to give you: ac
And you must find 2 numbers that add to give you: b
Common factor both brackets.
Lastly, group the factors.

So in this example 2x^2 + 9x + 4

- Multiple a and c value: 2 x 4 = 8

- Now find factors of 8 that equal to 9

8: 1 x 8 =8 also 1+8 = 9

2 x 4 = 8

- Two numbers that add to the b value are placed instead of the orginal b value.

2x^2 + 9x + 4

2x^2+ 8x+ x+ 4

( 2x^2 + 8x) (x+4)

- Common Factor the two brackets.

2x (x + 4) + 1( x +4)

- Here you just group them.

Solution : ( 2x+ 1) (x+ 4)

Example 2) 4m^2 - m + 7

First of all can you common factor it?

No.

So when you multiply ac it gives you product of 28:

You cannot find factors of 28 that equal to 1.

So it's NOT POSSIBLE.

When finding two correct numbers that give you the value ac and add up to give the value b. There are two methods to choose from:

Trial and Error and Decomposition.

In Trial and Error:

1st 2 numbers i nthe bracket are factors of 'a' value.
Last two numbers in the bracket are factors of 'c' value.

In Decomposition:

Step#1: Find two values that multiply and add to the 'b' value.
Step#2: With the two values break down the middle term in the equation.
Step#3: Group first two terms and second two terms.
Step#4: Common factor both.

Which method is dependant on you. However, knowing the methods will help you in the future.

Dealing with Complex Trinomial Work Problems.

1)A rectangle has area defined by 6x^2 + 11x - 35

Area is :

6x^2 + 11x - 35

a) Factor to find algebraic expressions for the length and width of the rectangle.

b) If x represents 8 cm, determine the perimeter and area of the rectangle.

Answer:

In the question an equation is given that represents the area of the rectangle. In part a) you are suppose to

a) 6x^2 + 11x - 35

= 6 x 35 = 210

210 = 10 x 21 = 210

21 + -10 = 11

= 6x^2 + 11x - 35

= 6x^2 + 21x - 10x - 35

= (6x^2 + 21x)+ (- 10x - 35)

= 3x ( 2x+ 7) -5 ( 2x + 7)

= (2x+7) (3x - 5)

b) Area: (2(8)+7) x (3(8) - 5)

= 23 x 19

= 437 cm^2

Perimeter: 2(2x+7) + 2(3x - 5)

= 2 (2(8)+7) + 2( 3(8) -5)

= 2(23)+ 2 (19)

= 46 + 38

= 84 cm

Therefore, the area of the rectangle is 437 cm^2 and the perimeter of the rectangle is 84 cm.

2) The height of a ball thrown from the top of a ladder can be appromixated by the formula h= -2t^2 + 4t + 48, where t is the time, in seconds, and h is the height, in meters.

a) Write the formula in factored form.

b) Determine when the ball will hit the ground.

a) h= -2t^2 + 4t + 48 can be common factored by 2.

= -2 ( t^2 - 2t - 24)

Break down the middle term.

= -2 (t^2 - 6t + 4t - 24)

Group the terms.

= -2 [ (t^2 - 6t) + (4t - 24)]

Common factor both.

=-2 [ t (t -6)+ 4 (t-6) ]

= -2 [ (t-6) (t +4) ]

= -2 ( t-6) (t+4) The equation is in factored form. [y= a (x-r) ( x-s)]

b) To find out when the ball will hit the ground.

You let the equation equal to 0.

0 = = -2 ( t-6) (t+4)

t-6=0

t = 6 seconds

t+4= 0

t=-4 seconds This seems unrealistic. Therefore, The ball will hit the ground at 6 seconds.