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Tonya hy Accelerated Integrated Algebra 1 8 3! |200q 2.1 Notes Add and Subtract Polynomials Vocabulary: 1, Monomial-A number, a variable, or the product of both which have whole nomber exponents. (Ex: 4, 5x, xyz) 2. Polynomial A monomial or a sum of monemials. Fach monomial is called a term (Ex: 922+2x46, has thee ferms.). 3. Degree of a polynomial - 7},. gveotest Aa vee of its 4 D (the exomple for #2 bas a deer af 2 4S Ferms, 4. Leading Coefficient - Tho cp offnj Bee aera peg decrease ‘ent fest ‘0 First term when the 5, Binomial Y or #2 has a leading coefficient rp fi | i A polynomia| with two terms. (Ex : Rx-8) 6. — Trinomial - A polynomial with three terms, (Ex + Bat 2a - 8) Example 1 Write the polynomial so that the exponents decrease from left to right. Identify the degree and the leading coefficient. Identify the number of terms. b) 4a + 5a* - 8a" c) -1le + 14c* + 18c° 5a4- 80% +4a Bee + Het -Ile Degree : 4 Degree : © heading Coefficient 5 heading Coefficient : 18 + of Terms: 3 Hof Terms : 3 Accelerated Integrated Algebra 1 2.1 Notes Add and Subtract Polynomials Example 2 Find the sum or difference. a) (2a? + 7) + (7a" + 4a - 3) b) (9b? - b + 8) + (4b? - b - 3) 2o% 474 'la* +4a-3 Gbt-b+8+4b2-b-3 fat +4a4H S3b7-2b +5 ¢) (7c? -6e+4)- (oe? - 5e? - c) d) (d? - 15d + 10) - (-12d? + 8d - 1) Ie5 Gc 4-974 5e%4¢ d*-|5d+10 412d? -8d+4 +H \3d*-23d +414 | Example 3 Write a polynomial that represents the perimeter of the figure (don't forget to combine like terms). 4x-3 axe ax+1 2x-14 Kt+2 2x+1 \Ox + 2 Gx-L QOS Ja-l+2e41+4x-34+x42 Fi \ fo uw Ye x oO 5x -| SZ 4x =D Accelerated Integrated Algebra 1 2.2 Notes Multiply Polynomials Example 2 Flower Bed You are designing a rectangular flower bed that you will border using brick pavers. The width of the border around the bed will be the same on every side, as shown. a. Write a polynomial that represents the total area of the flower bed and the border. b. Find the total area of the flower bed and border when the width of the border is 1.5 feet. a) Ye* +22n +30 s) 12 Example 3 Shipping A box used for shipping is shown at the right a. Write a polynomial that represents the area of the base (nea) of the box, inches b. Write a polynomial that represents the volume of hn + 2) the box. — c. What is the volume if the length of the shortest side ainenes is 8 inches? a) A= n= +2n b) Ven?+6n*48n Ton Ya he / Accelerated Integrated Algebra 1 42 | 2] 200 g 2.3 Notes Find Special Products of Polynomials Square of a a Binomial Pattern Algebra % SBraiiple @tbp=at2bt+ HP @+3pexrt be +9 (a- bea — 2b +h? (3x — 2)? = 9x7 — 12x + 4 Example 1 Find the product. as ay oc aA a => a) (%+, 99%, b) (32 4.7, c) (Zw 3h, d) (10r - 3s}h, P+lByt8l 24422449 Hw [2 +94 \00r?- Ors +45" k Sum and Difference Pattern Algebra Wsakpis (+ de-hac-F (x + 5)@ — 5) =? — 25 Example 2 «) (69) @*9) by (2 20) Ge + 20) at -8 | ve - 400 Tw y Wy c)(4-w) (4+w) d) (5r + 1) (5r - 1) *+16 Z5r*- 1 Factoring Polynomials in the Form of ax? +bx+c (6 STEPS) The “Six Steps” Method 1. Find the product of ac. 2. Find two numbers whose sum eguals b and whose product is ac. 3. Rewrite the polynomial, replacing bwith the two new terms. 4. Group, using parenthesis. 5. Factor out GCF from both parenthesis. 6. Rewrite as two binomials. 350% 4+ 252 +4x 420 4(5x24 25x) (4x +20) 5 5x (245)+4 (x+5) 6 (S244) (x45) 1 5x2 18x +9 15-94-45 Q1S53 3.5x27- 1S2- 3x + gq 4. (Se?-|Sx) (3x +9) 5.52 (x- 3) +3 (x -3) 652-3) (2-3) 3, 4n?- 15-25 4.4+-252 -100 2,-20,5 3, 4n2- 20n+5n -28 4 tn?-200) (45h -28) 5. 4nln- 5) +Sln-S, CantS)(n- 5 5 eke ~lIn+4 ) 44-4216 z,-lb,-! ned Yn2-lén-In + ; (itn? tbe) -40 +4) 5 in(n-d) -tln-) 6, (ant) (n-4) 6. (3x-5) (x41 4, 2b? +3b-9 1.2--9= -18 2. 6,-3 3, 267+ 65-3b-9 4, (2b2+ 6b)-3b-Y 5. 2b(b+3)-3(b+3) 6. (25-3) (b43) 6. Gx? + fix -10 2. IS, 3, bx2+ IS - 4x -10 4. (one IS 4x-10) 5, 3x (2x#9)-2(2x +8) 6. Gx-2(2x+5) Accelerated Integrated Algebra 1 2.5 Notes Solve Polynomial Equations in Factored Form Vocabulary: 1. and the other side is a product of polynomial factors. Roots - the solutions to an equation when one side of the equation is zero (Ex. 6x? + 12x = 0) 2. Vertical Motion Model - describes the height of a projectile. hz -l6t?+vt+s t=time in seconds vzinitial velocity in feet per second szinitial height in feet Example 1 Solve the equation. a) (m- 7)(m - 9)=0 b) (5n + 10)(4n + 12)=0 c) (2z - 8)(z + 5)=0 m- "TsO m-4=0 Sn+l0=0 4n+(2=0 Qz-3=0 zt5=0 +7147 +9 +49 “ID 10-12 “IZ W843 5 es m= 7 m= 49 Bnez-ib dns 12 222 8 Zi-5 SB A 2B answers ni-2 nee 3 zi4 Example 2 . q=0 q+!6=0 Solve the equation. = -I6 - lo qe 7 =+I6 a) q + 16q=0 aie b) 4k? - 8k = 0 2)e (4h) (k- k-Z=0 191 2)=- fh fy tebe t= 37 7 22 i 22 c) 12h? = 36h ih 36h. 6 - = + Accelerated Integrated Algebra 1 2.6 Notes Factor x? + bx +c When trying to factor trinomials in the form of x? + bx +c let x’ + bx +c = (x + p)(x + q) so that the following is true... p+q=b AND pqze Example 1 Factor the trinomial. a) x? + 10x + 16 b) z?-7z+ 12 c) 2? + 5z - 36 (x +8) (x+2) (z-2)(z-4) (z- 4) (z+9) Example 2 Solve the equation. a) x2+30= Ix b) 52-36 2)=4 Ie mie S-35-6=4 xrellx+30=0 ad at (2-6) (x-5)=0 87-35 -10=0 (s-5)(s+2)=0 z-6=0 | x-5=0 $-5=0 | 342-0 +@ +G@ +5 +5 +5 +5 72. “2 xz6© ri5 ; $25 s2-24 Accelerated Integrated Algebra 1 2.6 Notes Factor x* + bx +c Example 3 Patio Area A community center is building a patio area }x ft} 100 ft along two sides of its pool. The pool is rectangular with [ a a width of 50 feet and a length of 100 feet. The patio area will have the same width on each side of the pool. t a o a. Write a polynomial that represents the combined area | of the pool and the patio area. com +1502 + 5, O00O ia b. The combined area of the pool and patio area should be 8400 square feet. How wide should the patio area be? © =, @ (2 +100) (x +50) =.x*+150x + 5,000 (e- -26) (x4! me)= 0 © x? +1S0x +5,000 = 8,400 rae zm GD oth ——— = 8,400 8,400 x? +150x - 3400 = 0 Example 4 Area Rug You create your own area rug from a square piece of remnant carpeting, You cut 4 inches from the length and 3 inches from the width. The area of the resulting area rug is 1056 square inches. a. Write a polynomial that represents the area of your areatog. 2 = Joe 4 |Z . What was the perimeter of the original piece of remnant e erneing? @ (x-3) (x-4)= 2? - Tx 412 © x2-Te +12 = 1,056 -1,056 - 1,056 ee 26)=0 x? Toe!-'|,044 = O — 24 2 =36x4 = /H4 Accelerated Integrated Algebra 1 2.7 Notes Factor ax’ + bx +c Example 2 Factor the trinomial. a) 4x? - 8x - 5 b) 5s*- 185-8 (244) (22-5) (5s+2)(s-4) When a is negative: (Ex. -2x? + 11x - 12) e You can factor out a -1 at the beginning e Find the factors of 2 e Find the factors of 12 (make the factors both negative) « Find the possible factorizations . Find the middle terms (x — 1)Qx — 12) | —12x— 2x = — 14x | X 1,2 =12,-1 | (e- 122e- 1) | —x— 24x = —25x | XK 1,2 -2,-6 | (&-2)@x-6) | —6x—4x=—10x | X 1,2 -6,-2 | @-6)(2x- 2) | -2x- 12x = —14x | X 1,2 | -3,-4 | @-3@x-4) | —4x- 6x =—10e | x 1,2 =4,-3 | @-4)(Qx- 3) | —3x-8e= —Ilx | <— correct Example 3 Factor the trinomial a) -x® - 3x + 28 e -4s?+65+4 (+4) (x+7) ” Csi) (as+4) Accelerated Integrated Algebra 1 2.8 Notes Factor Special Products Difference of Two ‘Squares Pattern Algebra Example > Pf —b = (a+ b= b) 9° — 4 = 3x)? — 2? = Bx + 2)(3x — 2) Example 1 Factor the difference of two squares. a) x’ - 36 b) 25p* - 144 c) 80 - 125s" (2-6) (x+6) (5p-| i2) (Sp+l 2) can’+ factor Perfect Square Trinomial Pattern -| Example 2 _ | Factor perfect square trinomials. a) 16r? + 40rs + 25s? b) 36x? - 36xy + 9y” Gr+5s) 7 (6x-3y)* et Accelerated Integrated Algebra 1 2.9 Notes Factor Polynomials Completely Vocabulary: 1) Factor by Grouping - fairing “evs and tren lod’ ing tor factors. (this is acta ly Step H or 42 @ Step. me 2) Unfactorable - ) polynomial thot connot be w product of polynomials with integer coetticients. vd 3) Feetgring Completely produc: or on -When a polynomial is written as the actorable polynomials with integer coefficier tS Example 1 Factor out common binomial. | a) 11x(x - 8) + 3(x-8) b) g(15 + g) - (q+ 15) c) 12(d - 3) + 2d(d- 3) (\in-+3)(x-8) (q-1)(o+15 (2d+12)(d-3) = NH) 2(d+6) (d-3) Example 2 Factor by grouping. a) 9x3 + 9x? - 7x - 7 = b) 3 + Qty - 35x? - oxy = (Gx.3 +92) (-"la.-"1) = (on 355 2iy- Gx = 9x? (xt1)-1(x+44) = (1ox8-352*)(2iy-Gxy) = et Daet) Si (aa -8y (xe1)= (Sx? 3) 22-7) Accelerated Integrated Algebra 1 2.9 Notes Factor Polynomials Completely Example 3 Factor completely. a) 4x? - 36x b) — 2y° - 16y? + 32y . Ut (x?-Q) = 2y (y2-8y +16) = Hoe(x+3)(x-3) 2y (y-4)* Example 4 Solve a polynomial equation. a) 4x3 + 48x? + 144x = 0 b) Sr? + 15r = 20r? Use (x24 120 +36) =O 57-20r*415r
