“FACTORING”

THINK ABOUT THE MEANING OF THE WORD FACTOR. I THINK THAT FACTORS ARE NUMBERS THAT ARE MULTIPLIED TO FORM A PRODUCT!

THEN WHAT FACTORS COULD BE USED TO END UP WITH THE PRODUCT (5x + 5y) ? I MUST ADMIT THAT I REALLY HAVE NO IDEA! I THOUGHT FACTORS WERE LIKE 3 X 4 = 12!

I’m noticing that they both have a 5 . . . DON’T JUST THINK ABOUT NUMBERS – THINK ABOUT PROPERTIES THAT YOU COULD USE . . . OR USE BACKWARD. Hmmmm…. (5x + 5y) I’m noticing that they both have a 5 . . .

THINK ABOUT THE DISTRIBUTIVE PROPERTY USED BACKWARD. OH!!!! (5x + 5y) = 5(x + y)

HOW WOULD YOU FACTOR (6z + 6j) (6z + 6j) = 6(z + j)

HOW WOULD YOU FACTOR (7a – 7x) ? (7a – 7x) = 7(a - x)

TO CHECK YOUR ANSWER, USE THE DISTRIBUTIVE PROPERTY TO CHECK YOUR ANSWER, USE THE DISTRIBUTIVE PROPERTY. BOTH EXPRESSIONS MUST BE EQUIVALENT. SO IF I HAVE TO FACTOR (5x + 5y) AND I THINK THAT THE ANSWER IS 5(x + y), THEN I CAN USE THE DISTRIBUTIVE PROPERTY TO CHECK! 5(x + y) = 5x + 5y

MANY TIMES THE FACTOR IS A VARIABLE. WHAT IS THE COMMON FACTOR OF (3x + yx) ? THE COMMON FACTOR IS x. SO (3x + yx) = x(3 + y)

(3x + yx) = x(3 + y) THIS IS CALLED “FACTORING” BECAUSE YOU END UP WITH TWO FACTORS THAT CAN BE MULTIPLIED TO RESULT IN THE ORIGINAL PRODUCT. Oh! It’s like instead of 3 x 4 = 12 you start with 12 and end up with 3 x 4 !

TRY FACTORING (7a + ab) (7a + ab) = a(7 + b)

(xy + xz) (xy + xz) = x(y + z) TRY FACTORING (xy + xz) (xy + xz) = x(y + z)

THINK ABOUT BOTH NUMERICAL AND VARIABLE COMMON FACTORS. TRY FACTORING THIS ONE! (10xy + 5xz) THINK ABOUT BOTH NUMERICAL AND VARIABLE COMMON FACTORS. (10xy + 5xz) = 5x(2y + z)

THINK ABOUT BOTH NUMERICAL AND VARIABLE COMMON FACTORS. TRY FACTORING THIS ONE! (24ab + 16ac) THINK ABOUT BOTH NUMERICAL AND VARIABLE COMMON FACTORS. (24ab + 16ac) = 8a(3b + 2c)

THINK ABOUT BOTH NUMERICAL AND VARIABLE COMMON FACTORS. TRY FACTORING THIS ONE! (45a2b + 35ab) THINK ABOUT BOTH NUMERICAL AND VARIABLE COMMON FACTORS. (45a2b + 35ab) = 5ab(9a + 7)

NOW YOU ARE GOING TO LEARN HOW TO FACTOR A TRINOMIAL! BUT I FORGOT WHAT A TRINOMIAL IS. . .

THE PREFIX “TRI” ALWAYS MEANS 3, SO A TRINOMIAL HAS 3 TERMS. OH! LIKE (x2 + 7x + 12) IS A TRINOMIAL.

THINK ABOUT THE MEANING OF BINOMIAL. YOU WILL BE GIVEN A TRINOMIAL AND ASKED TO REWRITE IT AS THE PRODUCT OF TWO BINOMIALS. THINK ABOUT THE MEANING OF BINOMIAL. “BI” MEANS 2. A BINOMINAL IS AN EXPRESSION LIKE (x + 3).

(x2 + 7x + 12) (x + 3)(x + 4) (x + 3)(x + 4) x2 + 4x + 3x + 12 YOU CAN FACTOR THE TRINOMIAL (x2 + 7x + 12) INTO THE PRODUCT OF TWO BINOMIALS: (x + 3)(x + 4) SO IF I USE FOIL, ARE THESE TWO EXPRESSIONS EQUIVALENT? (x + 3)(x + 4) x2 + 4x + 3x + 12

BUT BE SMART ABOUT YOUR GUESSES! BASICALLY, YOU HAVE TO USE “GUESS-AND-CHECK” TO FACTOR A TRINOMIAL LIKE x2 + 7x + 12 BUT BE SMART ABOUT YOUR GUESSES! (x + 3)(x + 4)= x2 + 7x + 12 I NOTICED THAT 3 + 4 = 7 AND 3 x 4 = 12.

TO FACTOR x2 + 8x + 15 THINK ABOUT WHAT BINOMIALS YOU COULD “FOIL”. Hmmm . . . (__ + __ )(__ + __) x 3 x 5 x2 + 5x + 3x + 15

TO FACTOR x2 + 8x + 15 THINK ABOUT WHAT BINOMIALS YOU COULD “FOIL”. (x + 3)(x + 5) = x2 + 8x + 15 THAT WORKS!

TRY TO FACTOR x2 + 12x + 20 INTO THE PRODUCT OF TWO BINOMIALS. I HAVE TO USE GUESS-AND-CHECK, BUT I HAVE TO BE SMART ABOUT MY GUESSES. I’LL TRY 4 AND 5 SINCE THEY ARE FACTORS OF 20.

TRY TO FACTOR x2 + 12x + 20 INTO THE PRODUCT OF TWO BINOMIALS. (__ + __)(__ + __) x x 5 4 NOW I HAVE TO FOIL TO CHECK. x2 + 4x + 5x + 20 = x2 + 9x + 20

(__ + __)(__ + __) x x 10 2 x2 + 10x + 2x + 20 = x2 + 12x + 20 TRY TO FACTOR x2 + 12x + 20 INTO THE PRODUCT OF TWO BINOMIALS. TRY ANOTHER PAIR OF FACTORS OF 20! (__ + __)(__ + __) x x 10 2 NOW I HAVE TO FOIL TO CHECK. x2 + 10x + 2x + 20 = x2 + 12x + 20

TRY TO FACTOR x2 + 20x + 36 INTO THE PRODUCT OF TWO BINOMIALS. I HAVE TO BE SMART ABOUT MY GUESSES. THE FACTOR PAIRS FOR 36 ARE (1 x 36) (2 x 18) (3 x 12) (4 x 9) and (6 x 6).

The only pair that adds up to 20 is (2 x 18). TRY TO FACTOR x2 + 20x + 36 INTO THE PRODUCT OF TWO BINOMIALS. (1 x 36) (2 x 18) (3 x 12) (4 x 9) and (6 x 6) The only pair that adds up to 20 is (2 x 18).

TRY TO FACTOR x2 + 20x + 36 INTO THE PRODUCT OF TWO BINOMIALS. x2 + 2x + 18x + 36

PRACTICE FACTORING THESE TRINOMIALS INTO THE PRODCUT OF TWO BINOMIALS: x2 + 10x + 24 x2 + 17x + 30

x2 + 10x + 24 = (x + 6)(x + 4) x2 + 17x + 30 = (x + 15)(x + 2)

NOW IT’S TIME TO LEARN TO FACTOR MORE DIFFICULT EXPRESSIONS LIKE 6x2 + 19x + 10 WELL . . . 10 CAN ONLY BE FACTORED INTO (5 x 2) OR (10 x 1). 6 CAN BE FACTORED INTO (2 x 3) OR (6 x 1).

NOW IT’S TIME TO LEARN TO FACTOR MORE DIFFICULT EXPRESSIONS LIKE 6x2 + 19x + 10 (__ + __)(__ + __) 6x 5 1x 2 6x2 + 12x + 5x + 10 6x2 + 17x + 10 . . . Nope!

NOW IT’S TIME TO LEARN TO FACTOR MORE DIFFICULT EXPRESSIONS LIKE 6x2 + 19x + 10 (__ + __)(__ + __) 3x 5 2x 2 6x2 + 6x + 10x + 10 6x2 + 16x + 10 . . . Nope!

NOW IT’S TIME TO LEARN TO FACTOR MORE DIFFICULT EXPRESSIONS LIKE 6x2 + 19x + 10 (__ + __)(__ + __) 3x 2 2x 5 6x2 + 15x + 4x + 10 6x2 + 19x + 10 . . . Yes!

YOU JUST HAVE TO TRY OUT ALL OF THE POSSIBLE COMBINATIONS UNTIL YOU FIND THE RIGHT ONE. EASY FOR HIM TO SAY!

FACTOR THIS TRINOMIAL INTO THE PRODUCT OF TWO BINOMIALS. 10x2 + 25x + 15 10x2 + 25x + 15 = (2x + 3)(5x + 5)

FACTOR THIS TRINOMIAL INTO THE PRODUCT OF TWO BINOMIALS. 15x2 + 16x + 4 15x2 + 16x + 4 = (3x + 2)(5x + 2)

THINK ABOUT FACTORING THE DIFFERENCE OF TWO PERFECT SQUARES THINK ABOUT FACTORING THE DIFFERENCE OF TWO PERFECT SQUARES. IT SEEMED EASY, DIDN’T IT? I THOUGHT IT WAS EASY . . . (49x2 – 25) = (7x + 5)(7x – 5)

BUT SOMETIMES YOU HAVE TO CONTINUE FACTORING OR YOU ARE NOT REALLY FINISHED. CONSIDER THIS ONE: (x4 – 16) IT LOOKS EASY! (x4 – 16) = (x2 + 4)(x2 – 4)

OH! SO I STILL HAVE TO FACTOR THAT BINOMIAL. DON’T STOP TOO SOON!!!! (x4 – 16) = (x2 + 4)(x2 – 4) (x2 – 4) IS ANOTHER DIFFERENCE OF TWO PERFECT SQUARES!! OH! SO I STILL HAVE TO FACTOR THAT BINOMIAL. (x4 – 16) = (x2 + 4)(x2 – 4) = (x2 + 4)(x + 2)(x – 2) DONE!

TRY FACTORING THIS ONE: DON’T STOP FACTORING TOO SOON! (x4 – 81) DON’T STOP FACTORING TOO SOON! (x4 – 81) = (x2 + 9)(x2 – 9) = (x2 + 9)(x + 3)(x – 3) DONE!

TRY FACTORING THIS ONE: DON’T STOP FACTORING TOO SOON! (16x4 – 1) DON’T STOP FACTORING TOO SOON! (16x4 – 1) = (4x2 + 1)(4x2 – 1) = (4x2 + 1)(2x + 1)(2x – 1) DONE!