Polynomials and Division Theory


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1 Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the coefficients nd n N is the degree f(x) = Know tht polynomil function is of the form: n x + n x n + n x n + + n x + Know tht polynomil of degree 0 is constnt x + 0 Know tht polynomil of degree is liner expression Know tht polynomil of degree is qudrtic expression Know tht polynomil of degree is cuic expression Use the nested form of polynomil to clculte specific vlues Know the Reminder Theorem, nmely tht if polynomil f (x) is divided y (x h), then the reminder is f (h) Apply the Reminder Theorem y using synthetic division (or long division) to find the quotient nd reminder when dividing polynomil y liner fctor of the form (x h) Use the Reminder Theorem to find the quotient nd reminder when dividing polynomil y liner fctor of the form (x + ) Know tht root of polynomil f is numer x for which 0 Show tht given numer is root of polynomil Know the Fctor Theorem, nmely tht if f (h) = 0, then (x h) is fctor of f (x); lso, if (x h) is fctor of f (x), then f (h) = 0 Use the Fctor Theorem to find ll roots of polynomil Use the Fctor Theorem to fctorise polynomils such s: 6x 5x 7x + 6 x 7x + 9 x x + x x + x 7x + 4x + 4 x 8x + 5 x 5x M Ptel (August 0) St. Mchr Acdemy
2 Higher Checklist (Unit ) Higher Checklist (Unit ) x + 4x 5x Upon fctorising polynomil f, determine the set of x vlues for which f(x) > 0 or f(x) < 0 Given polynomil f, determine, y fctorising f(x) c, the set of x vlues for which f(x) = c, f(x) > c or f(x) < c (c R) Use the Fctor Theorem to find unknown coefficients in polynomil, for exmple, find k given tht (x + ) is fctor of x + x + kx + Determine the eqution of polynomil from knowledge of where the grph crosses the x  xis nd y  xis Sketch the grph of polynomil, indicting sttionry points nd intersections with xes Qudrtic Theory Skill Achieved? Know tht qudrtic expression is of the form: x + x + c Know tht qudrtic function is of the form: x + x + c Know tht qudrtic eqution is degree polynomil equted to zero: x + x + c = 0 Recognise the generl shpe of the grph of qudrtic function (prol): ( > 0, minimum) ( < 0, mximum) M Ptel (August 0) St. Mchr Acdemy
3 Higher Checklist (Unit ) Higher Checklist (Unit ) Know tht the roots (solutions) of qudrtic eqution re otined y solving: x + x + c = 0 Clculte the roots of qudrtic eqution y using the Qudrtic Formul: ± 4c x = Use the Qudrtic Formul to solve qudrtic, expressing the roots in exct (surd) form, i.e. P ± Q R Know tht there re 4 techniques for solving qudrtics: Using grph Fctoristion Completing the Squre Qudrtic Formul Know tht the discriminnt of qudrtic def = x + x + c is: D 4c Use the discriminnt to decide ( discriminte etween) the numer nd nture of solutions of qudrtic eqution x + x + c = 0: 4c > 0 4c = 0 ( rel nd distinct) ( rel nd repeted) 4c < 0 (no rel solutions) Know tht when the discriminnt is positive, the rel roots re either oth rtionl or oth irrtionl Solve prolems involving the discriminnt such s, find the vlue(s) of k for which x 5x + (k + 6) = 0 hs equl roots Use the discriminnt to show tht given qudrtic hs rel roots for ll vlues of vrile, for exmple, ( k) x 5kx k = 0 Given generic fetures of qudrtic function, such s the sign of the discriminnt nd, identify possile grph for the function Know tht if qudrtic eqution hs distinct roots nd, then it cn lwys e written s y = k (x ) (x ) Know tht to complete the squre in qudrtic expression M Ptel (August 0) St. Mchr Acdemy
4 Higher Checklist (Unit ) Higher Checklist (Unit ) mens writing the qudrtic in the form: y = (x + p) + q Know tht the process of completing the squre cn e used to rewrite ny qudrtic expression Complete the squre in qudrtic expressions such s: x + x 8 x + 6x + x 0x + x + 6 x + 4x x + 4x x + 8x + 7 9x 8 x 9 5x 4x + x Know tht qudrtic written in the form y = (x + p) symmetry xis eqution x = p nd turning point ( p, q ) + q hs Know tht if stright line intersects prol, then the liner nd qudrtic equtions re to e equted to mke new qudrtic eqution Use the discriminnt to decide whether line intersects prol, nd if so, how mny times, using the criteri: D > 0 D = 0 ( intersection points) ( intersection point  tngency) D < 0 (no intersection) Here, D is the discriminnt of the new qudrtic eqution Know tht clculting f (0) gives the y  intercept of prol M Ptel (August 0) 4 St. Mchr Acdemy
5 Higher Checklist (Unit ) Higher Checklist (Unit ) Sketch grphs of qudrtics functions, noting specil fetures such s mximum or minimum turning point, intersections with xes nd loction nd eqution of the symmetry xis Know tht qudrtic ineqution is qudrtic eqution with the equlity replced y n inequlity Solve qudrtic inequtions y sketching the grph of the corresponding qudrtic function Solve qudrtic inequtions such s: x x + < x x < 0 x + x 0 x + 4x > 0 x 5 0 x x 0 x + x > 0 Given the x intercept(s) nd the y intercept of prol, determine the eqution of the prol, especilly if the prol hs eqution px (x q) M Ptel (August 0) 5 St. Mchr Acdemy
6 Higher Checklist (Unit ) Higher Checklist (Unit ) Integrtion Skill Achieved? Understnd the concept of integrtion in wys: The opposite of differentition (ntidifferentition) Clculting res Know tht if there re functions F nd f such tht: F (x) then F is clled n ntiderivtive (k primitive) of f Know tht not ll functions hve ntiderivtives Know tht if function f hs n ntiderivtive F, then f is integrle nd F is lso clled n integrl of f Know tht if function is integrle, then it hs infinitely mny integrls Know tht the process of working out F is clled integrtion Know tht integrls re of types: Definite Indefinite Know the nottion for n indefinite integrl of f : F = f dx Know the Fundmentl Theorem of Clculus, nmely, tht if: F (x) then the definite integrl of f is worked out s, f (x) dx = F () F () ( x ) Know the result: n x dx = n + x n + + C ( n ) M Ptel (August 0) 6 St. Mchr Acdemy
7 Higher Checklist (Unit ) Higher Checklist (Unit ) where C is the constnt of integrtion Know the integrtion rules : n x dx = n x dx ( is constnt) (f (x) ± g (x ) dx = f (x) dx ± g (x) dx Know how to simplify rtionl expressions efore integrting Know tht the integrl of the zero function is constnt, i.e.: 0 dx = C Know tht the integrl of constnt function is liner function, i.e.: k dx = k dx = kx + C Know tht the integrl of liner function is qudrtic function, i.e.: A ( Ax + B) dx = x + Bx + C Know tht the integrl of qudrtic function is cuic function, i.e.: ( Ax Bx C A + + ) dx = Integrte functions such s: B x + x + Cx + D 0 x 4 x x 5 x x 5 M Ptel (August 0) 7 St. Mchr Acdemy
8 Higher Checklist (Unit ) Higher Checklist (Unit ) 4 x x x x 5 ( x ) ( x + ) x 4 x x Evlute definite integrls exctly, for exmple: x dx 0 x dx Given the vlue of definite integrl nd one of the limits, evlute the other limit, for exmple: t 4 sin ( x + 5 5) dx = 0 Know tht the region of integrtion is the set of x vlues over which function is to e integrted Know tht the re ounded (enclosed) y the curve y =f (x), the lines x = nd x = nd the x xis is given y the definite integrl: Are = f (x) dx (ove x xis: f (x) > 0, x [, ]) M Ptel (August 0) 8 St. Mchr Acdemy
9 Higher Checklist (Unit ) Higher Checklist (Unit ) Are = f (x) dx or (elow x xis: f (x) < 0, x [, ]) Know tht oth the ove res re positive Know tht for function tht is oth positive nd negtive over the region of integrtion, the intersection point(s) of the grph with the x xis must e known efore integrting Know tht for function tht is oth positive nd negtive over the region of integrtion, the totl re ounded the curve, the x  xis nd the integrtion limits is found y: Clculting ll res ove the x xis (ech of which is positive) Clculting ll res elow the x  xis (ech of which is negtive) Ignoring the negtive signs for ll negtive res Adding ll the (now positive) ove res Clculte the re ounded y the x xis, stright line nd given limits (one of which my hve to e found) Clculte the re enclosed y the x xis, prol nd limits (which my hve to e found) Clculte the re enclosed y the x xis, cuic grph nd limits (which my hve to e found) Know tht the re ounded (enclosed) y two curves is otined y integrting top function ottom function etween two limits nd : A = (f (x) g (x ) dx = f (x) dx g (x) dx where f (x) g (x) nd x Know tht to find the re enclosed etween curves, the intersection points of the two grphs must e known nd they re the limits of integrtion Find the re enclosed etween: A prol nd stright line A stright line nd cuic grph M Ptel (August 0) 9 St. Mchr Acdemy
10 Higher Checklist (Unit ) Higher Checklist (Unit ) A prol nd cuic grph prols Clculte the re enclosed etween other curves, for exmple: x nd g (x) = x x (x [0,]) Know the grphicl integrtion rules : f (x) dx = f (x) dx c f (x) dx + c f (x) dx = f (x) dx ( c ) Know tht (first order ordinry) differentil eqution is n eqution involving the first derivtive of function: dy dx = g (x) Know tht solving differentil eqution mens finding the originl function y y integrtion Know tht initil conditions re required to solve for the constnt of integrtion Solve differentil equtions such s: dy dx = 4x 6x when x = nd y = 9 M Ptel (August 0) 0 St. Mchr Acdemy
11 Higher Checklist (Unit ) Higher Checklist (Unit ) Addition Formule Skill Achieved? Know the ddition formule (k compound ngle formule): sin ( x ± y ) = sin x cos y ± cos x sin y cos ( x ± y ) = cos x cos y sin x sin y Use compound ngle formule to expnd expressions such s: sin ( F G ) cos ( 9 ) Use compound ngle formule in reverse to simplify nd find exct vlues of extended expressions such s: π cos 4 π cos 4 + π sin 4 π sin 4 sin 70 cos 40 cos 70 sin 40 Apply compound ngle formule to find exct vlues when given the exct vlues of sin x, cos y, cos x nd sin y Given rightngled tringle with sides nd one ngle x, find the exct vlues of sin x nd cos x Apply compound ngle formule to find exct vlues y decomposing ngles, for exmple: sin 7π = sin π π + 4 cos 5 = cos (45 0) Apply compound ngle formule to prove trigonometric identities such s: cos ( x + y ) cos ( x y ) = cos x cos y sin x sin y tn ( x + y ) = tn x + tn y tn x tn y sin ( A + B) + sin ( A B) = sin A cos B Know the Pythgoren Identity : M Ptel (August 0) St. Mchr Acdemy
12 Higher Checklist (Unit ) Higher Checklist (Unit ) sin x + cos x = Given n exct vlue for sin x, where x is n cute ngle, find the exct vlues of cos x nd tn x Given n exct vlue for cos x, where x is n cute ngle, find the exct vlues of sin x nd tn x Given n exct vlue for tn x, where x is n cute ngle, find the exct vlues of sin x nd cos x Know the doule ngle formule : sin x = sin x cos x cos x = cos x = cos x = sin x sin x Given n exct vlue for sin x, where x is cute, pply doule ngle formule to find the exct vlues of sin x nd cos x Given n exct vlue for cos x, where x is cute, pply doule ngle formule to find the exct vlues of sin x nd cos x Given n exct vlue for tn x, where x is cute, pply doule ngle formule to find the exct vlues of sin x nd cos x Rerrnge doule ngle formule in the form: sin x = ( cos x) cos x = ( + cos x) Know tht doule ngle formule re used to chnge etween terms of the form sin x nd sin x (nd similrly for cosine) Given n exct vlue for cos x with x cute, pply doule ngle formule to find the exct vlues of sin x nd cos x Apply doule ngle formule (nd possily) the Pythgoren Identity to solve trigonometric equtions such s: cos x + cos x = (x [0, 60]) sin x cos x = 0 (x [0, 80]) cos x + 0 cos x = 0 (x [0, π]) cos x 5 cos x 4 = 0 (x [0, π)) M Ptel (August 0) St. Mchr Acdemy
13 Higher Checklist (Unit ) Higher Checklist (Unit ) Use rerrngements of doule ngle formule to prove trigonometric identities such s: sin x + cos x = tn x 4 cos α 4 sin α = cos α cos x + cos x = tn x 4 sin x = (cos 4x 4 cos x + ) 8 Comine doule ngle formule with compound ngle formule to find expressions nd exct vlues for quntities such s, sin x, cos x, sin 4x nd cos 4x The Circle Skill Achieved? Know tht circle is the set of points in the plne ll of which re the sme distnce from given point; the given point is clled the centre nd the common distnce is clled the rdius Know tht the eqution of circle with centre ( 0, 0) nd rdius r is: x + y = r Given the eqution of circle in the ove form, write down the centre nd rdius Write down the eqution of circle of given rdius nd centre ( 0, 0) Know tht the eqution of circle with centre (, ) nd rdius r is: ( x ) + ( y ) = Write down the eqution of circle of given rdius nd centre ( ), r Given the eqution of circle in the ove form, write down the centre nd rdius Sketch circle of given rdius nd centre M Ptel (August 0) St. Mchr Acdemy
14 Higher Checklist (Unit ) Higher Checklist (Unit ) Know tht the expnded form (k generl eqution) of circle is: x + y + gx + fy + c = 0 where the centre is ( ) g, f nd the rdius is g + f c, ssuming tht g + f c > 0 Rerrnge circle eqution tht is not in expnded form to one tht is in expnded form Given the eqution of circle in expnded form, clculte the rdius nd centre Write down the eqution of circle in expnded form given the rdius nd centre Given missing vrile in the generl eqution, determine which vlues render the eqution tht of circle, for exmple, determine the vlues of k for which the following eqution represents circle: x + y + 4kx ky k = 0 Know the condition for when the expnded form does not represent circle, nmely: g + f c 0 Given the centre of circle nd point on it, determine the eqution of the circle Given the eqution of circle, determine whether or not point lies on the circle Decide whether point lies on, in or outside circle of given rdius Know tht line nd circle my intersect: At no points At exctly point (tngent) At points Know tht if line is to intersect circle, then the line eqution in the form y = mx + c is sustituted for y into the circle eqution, which upon simplifying produces qudrtic eqution Ax + Bx + C = 0. If this eqution hs no solutions ( B 4AC < 0) then the line does not intersect the circle M Ptel (August 0) 4 St. Mchr Acdemy
15 Higher Checklist (Unit ) Higher Checklist (Unit ) If this eqution hs exctly solution ( B 4AC = 0) then the line is tngent to the circle If this eqution hs distinct solutions ( B 4AC > 0) then the line crosses the circle in two plces In the ltter two cses, once n x  coordinte is otined, the y  coordinte is found y sustituting the x  coordinte into the eqution y = mx + c Find the intersection point of circle nd line tht is tngent to it Find the intersection points of circle nd line tht intersects it more thn once Find the equtions of the tngent lines to circle from given point outside the circle Find the eqution of tngent line to circle given point lying on the circle Know tht circumscried circle (k circumcircle) is circle tht psses through ll vertices of polygon Know tht every tringle hs circumscried circle Know tht the perpendiculr isectors of tringle meet t the circumcircle s centre Given the vertices of tringle, determine the eqution of the circle tht psses through these points Given the equtions of circles, clculte the sum of their rdii nd clculte the distnce etween their centres Know tht for circles: If the sum of their rdii is greter thn the centre  centre distnce, then the circles intersect t points (nd vice vers) If the sum of their rdii equls the centre  centre distnce, then the circles intersect t point (nd vice vers) If the sum of their rdii is less thn the centre  centre distnce, then the circles do not intersect (nd vice vers) Given the equtions of circles, determine whether or not they intersect M Ptel (August 0) 5 St. Mchr Acdemy
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