Quant · Algebra

Algebraic Identities and Polynomials

3 sub-topics · 29 lessons · 73 questions · 13 PYQs

Videos are being made — every lesson's notes and practice are ready now.

Algebraic Identities13 lessons · 36 questions · 4 PYQs

Polynomials and Remainder Theorem10 lessons · 27 questions · 8 PYQs

Maxima and Minima in Algebra6 lessons · 10 questions · 1 PYQs

Formula sheet

Square of a three-term sum
$(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(ab+bc+ca)$
Bounds for the pairwise sum
$a^{2}+b^{2}+c^{2}=k\Rightarrow -\dfrac{k}{2}\le ab+bc+ca\le k$
AM-GM inequality
$u,v>0:\ u+v\ge2\sqrt{uv};\quad \dfrac{u+v+w}{3}\ge\sqrt[3]{uvw}$
Completing the square
$ax^{2}+bx+c=a\left(x+\dfrac{b}{2a}\right)^{2}+c-\dfrac{b^{2}}{4a}$
Fixed sum, largest product
$x+y=S\ (x,y>0)\Rightarrow xy\le\dfrac{S^{2}}{4};\quad px+qy=S\Rightarrow xy\le\dfrac{S^{2}}{4pq}$
Squares and cubes from sum and product
$x^{2}+y^{2}=(x+y)^{2}-2xy,\quad x^{3}+y^{3}=(x+y)^{3}-3xy(x+y),\quad x^{3}-y^{3}=(x-y)^{3}+3xy(x-y)$
Two quartic factorisations
$x^{4}+x^{2}y^{2}+y^{4}=(x^{2}+xy+y^{2})(x^{2}-xy+y^{2}),\quad x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})$
Sum of three cubes
$a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)=\tfrac{1}{2}(a+b+c)\left[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}\right]$