In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case

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1 Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio of the for (), like ters re ters with the se power of the vribles (i this cse x d x d A. Fctorig: d x d p x re like ters. We lso reeber the properties of fctorig of the for: x y x y to siplify expressios. I geerl, expressio is sipler i its fctored for, but we y wt to first expd d the fctor bck i order to siplify expressios. Look t exples. to.6 i the textbook s pplictios of fctorig d expdig. A3. Properties of powers: Reeber the geerl properties of powers: ()... ties by defiitio. This produces the followig properties (eorize): (3) b b ; b (tke 0 i the first id.) b for y, br d, N, but (3) c be geerlized for, Z d for, Q. As exercise do Exple.7 A4. Frctios: reeber tht we eed to fid the l.c.. of the deoitors i order to dd frctios with differet deoitors. Do exples.8 to. i order to siplify frctios.

2 B. Lier equtios: B. Vribles i lier equtios: There re vrious istces i which expressio depeds o give vrible. A vrible is qutity which is free to chge its vlue. For exple, the su of the iterior gles of polygo with sides (d vertices) s below: is (3) S ( ) 80 o (becuse polygo with sides c be split ito djcet trigles). I (3) S is the expressio which depeds o the vrible. See exple.3 to the ed. A equtio which is true for y vlue of the vrible sy x is clled idetity. Lier equtios re typiclly solved by usig properties of (rel) ubers: we c dd d subtrct by y uber d we c ultiply d divide with y ozero uber d the equtio reis the se. Note: Never ultiply d divide by 0! Surprisig results c be foud if you do, such s = s result of b. Do probles fro exercises set A d B. C. Chgig the subject i forul: A expressio which depeds o oe vribles or ore c be solved with respect to y of its vribles. For bh exple, if A (s i the re of trpezoid with bses d b d height h), the we c solve for i ters of A, b d h. Through this process, we sy tht we ke the subject of this forul. See the other exples d exercise set C.

3 D. Qudrtic fuctios : A qudrtic fuctio is fuctio of the for (4) x b x c with, b, cr. The correspodig equtio: (5) x b x c 0 is clled qudrtic equtio. We will study the followig topics for qudrtic fuctios d equtios: D. Wys to solve qudrtic equtio of the for (5): ) Copletig the squre which lso produces the qudrtic forul b) Fctor the qudrtic fuctio d solve the equtio fterwrds D. Grphs of qudrtic fuctios. D. ) Copletig the squre d the qudrtic forul: Note tht (6) b c b b c b b 4 c x b x c x x x x 4 4 Therefore, if we let (7) b 4c (which is clled the discriit of the qudrtic), we hve tht (8) b x. 4 The techique outlied bove is clled copletig the squre, d is very useful for solvig qudrtic equtios d for grphig qudrtic fuctios. Exple: Execise E : Proble, prts i) d iv). Foul (7) bove produces the qudrtic foruls, which is the forul for the roots of the qudrtic equtio x b x c 0. We distiguish 3 cses: i) If 0,we hve tht (9) b x, (two distict rel roots for positive discriit) b x, ( oe repeted or double root for zero discriit), the qudrtic equtio (5) does ot hve y rel roots. ii) If 0,we hve tht (0) iii) If 0 Meorize foruls (7), (9) d (0) d use these to solve fro Exercise set F : i), vi) i) vi) d 3. 3

4 b) Qudrtic equtios of the for (5) could lso be solved by fctorig the qudrtic fuctio directly. d x (sice x re the roots, the this expressio should hve s fctors x x, or verify () directly. First, ote tht () x b x c x x x x x d x () produces Viete s reltios, which re useful for fctorig or other types of probles: b x x (). These reltios re useful for obtiig the roots (d therefore the fctorig () fro c x x the coefficiets, b d c. They re especilly siple for the cse. Optiol: Note: It is soeties esier to fid first y x d y x for which Viete s reltios () becoe: (3) y y b y y c For, the fctorig ethod is to fid first y d y usig (3), d the fctorig () becoes x y x y x b x c. For, we fid y d y usig (3), d the fctorig is: ( x) x y y b x c x x f Ed Optiol Exples: Follow Exples.3,.4 d.5 the fro Exercise set D do probles 3 (choose 3 poits), 4 (choose 3 poits), 7 iv d v, 8 v d vi, 9 v d vi d. 4

5 D. Grphs of qudrtic fuctio. To grph qudrtic fuctio give by: (4) b squre (8) x. 4 x b x c, we use the for obtied fter copletig the Optiol: Before obtiig properties of the grph (which we ll ke us ble to grph fuctio of the for (8)) ote tht: ) The grph of x is: Also, if we kow the grph of the fudetl fuctio be obtied by trsfortios of grphs. Reeber tht: x s bove, the grph of the geerl qudrtic (8) c The grph of f ( x A) is horizotl shift to the right by A uits (for A 0) of the grph of f (x) ; The grph of B is verticl shift dow by B uits (for B>0) of the grph of f (x) ; The grph of f (x) is verticl stretch (for >) or shrik (for 0<<) ties of the grph of f (x). Ed Optiol The properties etioed bove (or direct observtio) llows us to sy tht the grph of the qudrtic fuctio f (x) give by (8) is prbol with : b The vertex t V, 4 ; Cocve up ( U ) for 0 d cocve dow for 0 ; b A xis of syetry of equtio: x ; y itercept t 0,c (the grph goes through the poit 0,c ) ; If 0, it hs two x itercepts t x,0 d x,0, if 0 0 it does ot hve y x-itercepts (the grph does ot cross the x- xis for 0 ) it hs oly oe itercept t It is useful to reeber (eorize) these 6 properties whe tteptig to grph qudrtic fuctio. x d if,,0 5

6 Exple: Fro Exercise set E do probles i, vii d x, d 4 i, ii d iii. 6