Trinomials+where+A+does+not+Equal+One


 * Trinomials where A does not equal 1 **

**‍ ****What does that mean? ** A trinomial in standard form is represented as y = ax2 + bx + cTo qualify as a trinomial the expression needs A quadratic term (ax2) A linear term (bx) A constant (c) If a (the coefficient in front of x2) is greater or less than 1 try.... **‍ ****How to solve the trinomial where A does not equal one **

First try to solve the trinomial by

Factoring out the Greatest Common Factoror Solve using the Perfect Square Trinomial Method

If neither of these apply we can use the Reverse FOIL Methods demonstrated. There are two different options.

Option 1: Guess and Check Works best when A is a prime number We will use **y=3x2+25x+28 ** as an example **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">(3x+ ) (x+ ) ** <span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 13px;">Because 3x and x are the only two terms whose product is 3x2 <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">2. Because all terms are positive the sign are all addition signs <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">3. Find two factors of 28 which, when FOILed complete the expression **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">2 and 14 are factors of 28 How to FOIL <span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 16px;"> **
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Find the first two terms of the two binomials

<span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 16px;">Put the numbers into the expression

**<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">(3x+2)(x+14) **

<span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 16px;">Find the middle term by multiplying the Inners and outer of FOIL

Add them together
**<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">2x+42x=44x ** <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">44x is not the middle term we are looking for so we need to try again

**<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">4 and 7 are factors of 28 **

<span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 16px;">Put the numbers into the expression

**<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">(3x+4)(x+7) **

<span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 16px;">Find the middle term by multiplying the Inners and outer of FOIL

<span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 13px;">This was the term we were looking for, so the answer is,

**<span style="font-family: 'Arial','sans-serif'; font-size: 21px;">(3x+4) (x+7) **

**<span style="font-family: 'Arial','sans-serif'; font-size: 21px;">Additional Example **

**<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">2x2 -4x -6 ** <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Set up two binomials **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">(2x )(x ) ** <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Find factors of -6 **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">2 and -3 ** <span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 16px;">Put the numbers into the expression **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">(2x-3)(x+2) ** <span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 16px;">Find the middle term by multiplying the Inners and outer of FOIL

**<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">-3x+4x=x **
<span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">That's not what we are looking for, <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Try again, rearrange the numbers, put them into the equation

**<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">(2x+2)(x-3) ** Find the middle term by multiplying the Inners and outers of FOIL **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">2x*-3 and 2*x ** <span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 16px;">Add them together **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">2x+-6x=-4x **

**<span style="font-family: 'Arial','sans-serif'; font-size: 21px;">Everything works! (2x+2)(x-3) **

<span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 21px;">Option 2: Factor by grouping <span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 13px;">Works best when A is not a prime number <span style="font-family: 'arial','sans-serif'; font-size: 13px;">We will use **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">4x2+16x+5 **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;"> as an example **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">4*15=60 **
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Multiply a and c
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Find all factors of 60 and their sums
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">Factors ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">1,60 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">2,30 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">3,20 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">4,15 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">5,12 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">6,10 ** ||
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">Sums ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">61 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">32 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">23 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">19 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">17 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">16 ** ||

**<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">10 and 6 work ** **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">4x2+6x+10x+15 ** **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">(4x2+6x) (10x+15) ** **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">2x(2x+3) 5(2x+3) ** = **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">(2x+3) (2x+5) ** =
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Look for the numbers whose sum equals the coefficient of the middle term
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Divide the middle term by the two factors
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Separate the first two and last two terms.
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Factor out GCF
 * 1) =<span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Rewrite the outside numbers as a second binomial =

**<span style="font-family: 'Arial','sans-serif'; font-size: 21px;">Additional Problem **

**<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Factor 8x2+15x+7 ** **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">8*7=56 **
 * 1) <span style="color: #3367d1; font-family: 'arial','sans-serif'; font-size: 13px;">Multiply a and c
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Find all factors of 56 and their sums
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">Factors ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">1,56 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">2,28 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">4,14 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">7,8 ** ||
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">Sums ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">57 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">30 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">18 ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">15 ** ||

**<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">7 and 8 work ** **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">8x2+8x+7x+7 ** **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">(8x2+8x) (7x+7) ** **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">8x(x+1) 7(x+1) ** **<span style="font-family: 'Arial','sans-serif'; font-size: 13px;">(8x+7) (x+1) **
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Look for the numbers whose sum equals the coefficient of the middle term
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Divide the middle term by the two factors
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Separate the first two and last two terms.
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Factor out GCF
 * 1) <span style="color: #3367d1; font-family: 'Arial','sans-serif'; font-size: 13px;">Rewrite the outside numbers as a second binomial

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