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SUMMARY An algebraic expression f(x) of the form f(x)= a0 + a1x +a2x2+…….+an xn ,where a0,a1, a2……. an are real numbers and all the index of x are non negative integers is called polynomial in x and the highest index n is called as the degree of polynomial. IMPORTANT FORMULAE (x+a)2 =x2 +2ax+a2 (x-a)2 =x2 – 2ax +a2 x2-a2 =(x+a) (x-a) x3+a3=(x+a) (x2-ax+a2)=(x+a)3 –3xa(x+a) x3-a3=(x-a) (x2+ax+a2)=(x+a)3 +3xa(x-a) (a + b + c) 2=a2+b2+c2+2ab+2bc+2ac (a + b) 3= a3+b3+3ab (a + b) (a-b) 3= a3-b3-3ab (a-b) a3+b3+c3-3abc = (a +b +c) (a2+b2+c2-ab-bc-ca) SPECIAL CASE : If a+ b +c=0 then a3+b3+c3=3abc ZEROS OR ROOTS OF A POLYNOMIAL A real number ‘a’ is a zero of a polynomial f(x),if f(a)=0.Here ‘a’ is called a root of the equation f(x) = 0 FACTOR THEOREM Let p(x) be a polynomial of degree greater than or equal to ‘a’, be a real number such that p(a)=0,then (x-a) is a factor of p(x),then p(a)=0. REMAINDER THEOREM Let p(x) be any polynomial of degree greater than or equal to one and ‘a’ be any real number. If p(x)is a dividend by (x-a),then remainder is equal to p(a).let q(x) be the quotient and r(x) be the remainder when p(x) is divided by (x-a),then Dividend=Divisor x Quotient + Remainder OBJECTIVE 1) If 4x4-3x3-3x2+x+7 is divided by 1-2x then remainder will be: 57 −59 55 −55 a) 8 b) 8 c) 8 d) 8 2) A quadratic polynomial is exactly divisible by(x+1) and (x+2) and leaves the remainder 4 after division by (x+3) then that polynomial is: a) x2+6x+4 b) 2x2+6x+4 c) 2x2+6x-4 d) x2+6x-4 3)
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