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are bis the sum of two numbers ps and qr, whose product, is (ps )(qr) = (pr)(qs) = ae., , Thus, to factorise ax? + by +, write bas the sum of two, Aumbers, whose product is ae., , By Using Factor Theorem, , Write the given polynomial p(x) = ax? +by+e, , n the form, , p(x) =a (" + b xt ‘)- ag(x) (i), , a a, , ae, , shere, og(x)ax2 +24 ©, aa, .e. firstly make the coefficient of x? equal to one if it is not, one., ind all the possible factors of constant term (<} of g(x)., a, , sing trial method, find the factors at which g(x) = 0 say,, r= 0 and x = 8. Further, write g(x) as the product of factors,, then g(x) =(x-a)(x—-B ) and put this value of g(x) in Eq. (i) to, pet required factors of p(x)., , actorisation of a Cubic Polynomial, , To factorise a cubic polynomial, we use the following steps:, Step 1 Write the given cubic polynomial, p(x) = ax? + bx” + cx +d in the form, 3b cd ;, pts) = af 09 4x7 +o x4— |= ag(x) (i), a a oa, , b cd, where, g(x) =x° +x? +2x4+—, , a aoa, i.e. first make the coefficient of x? equal to one if it, is not one and then find the constant term., , d, , Step II Find all the possible factors of constant term (4) of, a, , g(x)., , Step U1 Check at which factor of constant term, p(x) is zero, by using trial method and get one factor of p(x),, (i.e. x — Q)., , Step IV Write p(x) as the product of this factor anda, quadratic polynomial,, ic. p(x) =(x - @) (ayx? +bx+c,), , Step V_ Apply splitting the middle term method or factor, theorem in quadratic polynomial to get another two, factors. Thus, we get all the three factors of given, , cubic polynomial., , Algebraic Identities, , An identity is an equality, which is true for all values of its, variables in the equality, ic. an identity is a universal truth., Some useful algebraic identities are given below:, (i) (x + y)” =x? + Quy + y?, (i) (x—y)? =x? — Sey +", (iii) x? —y? =(x-y)(x+y), (iv) (x +.a)(x+b) =x" +(a + b)x + ab, (v) (xty +z)" =x? ty? +27 + Qry + Qyz + Bex, = Ex? + Wry, (vi) (x+y)? =x? + y> + 3xy(x + y), =x ty? + 3x7y + 3xy?, (vii) (x—y)® =x° —y? — 3xy (x -y), =x -y?- 3x7y + 3xy”, (viii) x — y® =(x—y)(x® +2y ty?) =(x—-y)[(x—y)” + 3xy], (ix) x9 + y3 =(x t+ y)(x? —xy + y?)=(x+y)[(x+ y)" — 3xy], (x) x3 +y% +.2° —3xyz, =(x+ytz)(x7 +y? +27 —xy —yz— 2x), =(xty+z)[(x+y +z)? —3(xy + yz +2x)], if (x+y +s) =Othenx? +y°? +3° =3xyz, , (xi) x* +y? +27 —xy —yz—2x, , = Sl(x—y)? +(y-2)? +=)"