NAME DATE PERIOD ALGEBRA FACTORING PRACTICE PACKET with notes FACTORING THE GCF THINK: Is there a number andlor variable that goes into every term? If so, single set of ( ) with the GCF outside. 1. FACTOR: 36psr - 24 nt, + 60p2r - 42r+, (rp*, ({rf -*p + id -1p) DISTRIBUTE BACK ro GHECK: ;?/ b -r ( to1r**p, 4fl = tc #bf'e _,{ _^Uf",. tt \ir_ y/l* r FACTORING THE DIFFERENCE OF 2 SQUARES THINK: DOES IT HAVE A GCF? If not, is it a binomial? Is it a DIFFERENCE (subtraction)? Are both terms perfect squares? If so, (^/ir/ + "lna)("./ir/ - Jr;A) 2. FACTOR: m2-4 ttt--+ ( rltzyrlr->) \ F'IL BACK ro cfieck: ff *,twln ' + -? tfr'+ FACTORING THE DIFFERENCB OF 2 SQUARES THAT CONTINUES TO FACTOR AS THE DIFFERENCE OF 2 SQUARES: THINK: Same questions as last one, but this time the second ( ) will keep going. Make sure you keep going until the DIFFERENCE is no longer the difference of two squares! 3. FACTOR: m8 -l (r,.+ *,) (;n 4 : D \ irnu*,j(r', ),(*=- i) --> (rv,u n t){nn'-,:)fu-)('--i) \ F,IL BACK ro crmc"''\ r^/orl W\ WXL r-! I 'y ffi)\6hdu) e^svqst Lb,
FACTORING THE DIFFERENCE OF 2 SQUARES WITH A GCF: THINK: Does it have a GCF? Yes! Factor that out FIRST with a SINGLE ( ). Then factor as the difference of two squares. 4. FACTOR: 80mr - 45mu W; (t(u-rr-q,trvtr (+oo3) (q*4) FOIL BACK FIRST TI#N DISTRIBUTE TI{E GCF TO CFIECK:,-rhrd/-,t/ \^rdh FACTORING PERFECT TRINOMIAL SQUARES THINK: DOES IT HAVE A GCF? If no, is it a TRINOMIAL SQUARE? Check thar BOTH the first and last terms are positive. Check that they are also perfect squares. If so, check to see if the middle term is corect for being a Trinomial Square: Take the square root of each of the end terms. Multiply. Double. Is that the same number as the middle term? If so, ONE ( ) with the power of 2 on the outside ( )'. (J1s/ I "l2nd)' Th" sign in the middle is the same as the sign of the Znd term of the Trinomial Square! 5. FACTOR: 49y'- 70y +25 ( - -- \'2^ \'t3 - h ) FoIL BACK ro CFIECK: twwr\\,u*fr* -/ Check the middle term: FACTORING PERFECT TRINOMIAL SQUARES WITH A GCF THINK: DOES IT HAVE A GCF? If yes, GCF( ) first, then factor as a Trinomial Square. 6. FACToR: 100$+120y +36 a +Gs{'* vtl+0 +(;') r3y FOIL BACK FIRST, THEN DISTRIBUTE TFM GCF TO CFIECK: AMcI\ qor Mrv[,--t, (
FACTORING A TRINOMIAL WITH A POSITIVE LAST TERM and NEGATIVE THINK: DOES IT I-IAVE A GCF? If not, is it a trinomial? If so, is it a TRINOMIAL SQUARE? If not, you'll need ( X ) if it's going to factor. Does ithave apositive LAST TERM? If so, you know that they'll both be the SAME SIGN. Now look at the sign of 2nd (or middle) term. In this case it's NEGATIVE. So both signs will be NEGATIVE. Now do the PRODUCT/SUM. You'll need 2 factors that MULTIPLY to the LAST TERM and ADD to 7. FACTOR: c2-l3c+36 (c q)cc -,.t) FOIL BACK TO CFIECK: FACTORING A TRINOMIAL WITH A POSITIVE LAST TERM and POSITTVE THINK: DOES IT HAVE A GCF? If not, is it a trinomial? If so, is it a TRINOMIAL I SQUARE? If not, you'll need ( X ) if it's going to factor. Does it have a POSITIVE LAST TERM? If so, you know that they'll both be the SAME SIGN. Now look at the sign of 2nd (or middle) term. In this case it's POSITIVE. So both signs will be POSITIVE. Now do the PRODUCT/SUM. You'Il need 2 factors that MULTIPLY to the LAST TERM and ADD to 8. FACTOR: C+l5c+56 G*r)(ctr) FOIL BACK TO CFIECK:
FACTORING A TRINOMIAL WITH A NEGATIVE LAST TERM and NEGATIVE THINK: DOES IT HAVE A GCF? If not, is it a trinomial? If so, is it a TRINOMIAL SQUARE? If not, you'll need ( X ) if it's going to factor. Does it have a NEGATIVE LAST TERM? If so, you know that they'll both be the DIFFERENT SIGNS. Now look at the sign of 2nd (or middle) term. In this case it's NEGATIVE. So one sign will be NEGATIVE and the other sign will be POSITIVE. I like to put the sign of the middle term in the first ( ), but you don't have to. That way I know that the factor in the first ( ) will always be the BIGGER factor (it's the sign that "wins" :) Now do the PRODUCT/SUM. You'll need 2 factors that MULTIPLY to the LAST TERM just like before, but this time the two factors need to SUBTRACT to the middle term. Why? Because when you add different signs, you don't actually add, you subtract! Then you take the bigger number's sign (the sign I'm putting in the first set of parentheses). 9. FACTOR: c2-5c - 36 (c -6&.*) ForL BACK ro check:,r@ar*l'\* wo(tt- FACTORING A TRINOMIAL WITH A NEGATIVE LAST TERM and POSITIVE THINK: DOES IT HAVE A GCF? If not, is it a trinomial? If so, is it a TRINOMIAL SQUARE? If not, you'll need ( X ) if it's going to factor. Does it have a NEGATIVE LAST TERM? If so, you know that they'll both be the DIFFERENT SIGNS. Now look at the sign of 2nd (or middle) term. In this case it's POSITIVE. So one sign will be POSITIVE and the other sign will be NEGATIVE. Now do the PRODUCT/SUM. You'[ need 2 factors that MULTIPLY to the LAST TERM just like before and SUBTRACT to the middle term. 10. FACTOR: a2 +26a - 56 [^nr$a-, FOIL BACK TO CHECK: Ct^A C,ITIEu,iI'I wowl futr I
FACTORING BY GROUPING WITH A + IN THE MIDDLE: THINK: Is there a GCF of ALL TERMS? If not, are there 4 terms? Make sure that they're in descending order first! Is there a negative sign in the middle (sign of the 3rd term)? If not, pair off the firstz terms and the second 2 terms with parentheses. YOU SHOULD ALWAYS HAVE A + SIGN BETWEEN TI{E 2 ( ) + ( ). Now factor out the GCF of each pair. What's left inside each ( ) SHOULD BE THE SAME BINOMIALS! (If not, it's not factorable.) This common binomial is now your new GCF of both pairs. Bring that GCF forward in one set of parentheses. Put whatever is left i_n a second set o{trlarentheses. 6. 11. FACTOR: (rto'-z4py(op' - 16 ) T( {p=-r) r'2(r,(,'-l) (rr"-ryrir4 FACTORING BY GROUPING WITH A - IN THE MIDDLE: THINK: Same as ex Imple above, but there is a negative sign in the middle (sign of the 3rd term). Make sure you + - first! YOU SHOULD ALWAYS HAVE A + SIGN BETWEEN THE 2 ( ) + ( ).Then follow instructions above. 12. FACTOR: 20gt - 4g' - 25g + 5 (*3'-qil*6rrt "s) +"ql( g ) -r (;S-D FACTORING BY GROUPING WHEN TIIERE IS NOTHING TO FACTOR OUT OF THE SECOND PAIR: Same as the 2 examples above, but there is nothing to factor out of the 2ndpair. PUT A " I " IN FRONT OF THE 2ND PARENTHESES! 13. FACTOR: 623 +322 +22 +l &*(n+\+ tq-t*d @**)(t{*d (;Ei(+1'il
TYPE OF FACTORING: Use either guess and check on X BOX I - rrarr COEFFICIENT TRINOMIAL THINK: Multiply the end numbers. Find 2 factors that multiply to this product and either add or subtract (depends on the sign of the last term) to the middle term. Bring down both of the end terms exactly as they are. Replace the middle term with the 2 factors you found. Make sure the signs work! When you add those 2 factors together you need to get the same number and sign as the middle term. Now factor by grouping. 14. l5x'+2x-24+ - 24 Tx'b)(u+ +) *4 lsx' +zy"- Aq ok [ -*rl?w itx'rldx*l{ x'aq ^ W" 6*+?Dl -(tn : *r3,,]1,. Uu' TYPE OF FACTORTNG: MULTIPLE TYPES OF FACTORING THINK: AM I DONE???? at each step! 15. 3xYr -3xzt6 \ 3xhr- /) l{ aq Tx'" 6t--J G x++)(q*-b) ^ii- *z)(tr r? +) =r-4 or4({rz\!"i-z ) F vx[$u* n$.2)h*4(q- 7)