Section 1.6 Factoring Trinomials Pages 28 - 31

What are trinomials? A trinomial will have 3 terms There are four steps used to factor each of them Check for a common monomial factor Look for special cases Factor by trial and error Continue factoring as long as possible

Check for a common factor For each trinomial, it simplifies the factoring process if you look for a common factor What is the common factor in this expression? x2 – x + Since each term is divisible by 2, the 2 is common Each term also contains a y, so the common monomial is 2y When factored: 2y ( x2 – 4x + 4 ) 2 y 8 y 8 y

What is the common factor? find the common factor, then check your answer 3x2y3 – 6x5y + 12x3z 3x2 14a2b2c2 + 7a3b3c3d + 21ab2c 7ab2c answer: answer:

Check for special cases Once all of the common factors have been removed, look for special cases The only special case we’ll look at here is the perfect square trinomial a2 + 2ab + b2

2. Check for special cases If there is a perfect square, the first and last terms must be perfect squares x2 + 4x + 4 The first term ( x2 ) and last term ( 4 ) are both perfect squares

2. Check for special cases a perfect square will also need to have a specific middle term take the square root of the first term, multiply it by the square root of the last term, then multiply that by 2

2. Check for special cases x2 + 4x + 4 in this equation, the square root of the first term is x the square root of the last term is 2 when multiplied, x ( 2 ) ( 2 ) = 4x since this matches the middle term in the original expression, this is a perfect square

2. Check for special cases x2 + 4x + 4 the factors for this will follow the pattern: ( x + 2 ) 2 the first number inside of the brackets is the square root of the first term, the second number is the square root of the last term the sign inside of the brackets will match the sign of the second term

Which of these are perfect squares? 4x2 + 16x + 25 8x2 + 16x + 9 x2 - 12x + 36 81x2 + 38x + 100 9x2 + 25x + 4 16x2 + 32x + 3 2x2 + 10x + 25 4x2 + 12x - 9

3. factor by trial and error if the expression is still not factored completely, you must factor by trial and error

3. factor by trial and error 3x2 - 10x + 8 to factor, you need all of the factors of the first and last terms first term: 1 x 3 last 1 x 8 , 2 x 4 these need to be combined until the correct set of factors is obtained

3. factor by trial and error 3x2 - 10x + 8 remember, if the last term is positive, both of the factors have the same sign if the second term is negative, then both are negative if it is positive, then both are positive if it is negative, then they have opposite signs

3. factor by trial and error 3x2 - 10x + 8 since there are only 2 factors for the first term, we can easily guess the first part of the 2 factors ( 3x ) ( x ) we also know that the signs inside of the brackets are negative, since the sign of the second term was negative, but the third term was positive - -

3. factor by trial and error 3x2 - 10x + 8 ( 3x - ) ( x - ) Now we must choose two factors of 8 and see if we obtain the correct answer This might take a few attempts.

3. factor by trial and error 3x2 - 10x + 8 ( 3x - 1 ) ( x - 8 ) When multiplied, we get: 3x2 - 24x - x + 8 This is not the correct answer.

3. factor by trial and error 3x2 - 10x + 8 ( 3x - ) ( x - ) We don’t need to do all of the steps of FOIL, we only need the middle term, so we only multiply the outside and inside terms

3. factor by trial and error 3x2 - 10x + 8 ( 3x - ) ( x - ) Lets try two different factors. When multiplied, we get: -6x - 4x Combined together, we have -10x These are now the correct factors 4 2

4. Continue factoring until reach factor is prime Remember to apply as many steps as needed until the trinomial is factored as much as possible