⭐⭐ formula ⭐⭐
★polynomial ★
Answers
(x + y)^2 = x^2 + 2xy + y^2
(x - y)^2 = x^2 - 2xy + y^2
Example 1: If x = 10, y = 5a
(10 + 5a)2 = 10^2 + 2·10·5a + (5a)2 = 100 +100a + 25a^2
The opposite is also true:
25 + 20a + 4a2 = 52 + 2·2·5 + (2a)2 = (5 + 2a)2
Consequences of the above formulas:
(-x + y)^2 = (y - x)^2 = y^2 - 2xy + x^2
(-x - y)^2 = (-(x + y))^2 = (x + y)^2 = x^2 + 2xy + y^2
Formulas for 3rd degree:
(x + y)^3 = x^3 + 3x2y + 3xy2 + y^3
(x - y)^3 = x^3 - 3x2y + 3xy2 - y^3
Example: (1 + a2)^3 = 1^3 + 3.12.a2 + 3.1.(a2)2 + (a2)^3 = 1 + 3a^2 + 3a^4 + a^6
(x + y + z)2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz
(x - y - z)2 = x^2 + y^2 + z^2 - 2xy - 2xz + 2yz
Factor Rules
x^2 - y^2 = (x - y)(x + y)
x^2 + y^2 = (x + y)2 - 2xy
or
x^2 + y^2 = (x - y)2 + 2xy
Example: 9a2 - 25b2 = (3a)^2 - (5b)^2 = (3a - 5b)(3a + 5b)
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
x^3 + y^3 = (x + y)(x2 - xy + y2)
If n is a natural number
x^n - y^n = (x - y)(x^n-1 + x^n-2y +...+ y^n-2x + y^n-1)
If n is even (n = 2k)
x^n + y^n = (x + y)(x^n-1 - x^n-2y +...+ y^n-2x - y^n-1)
If n is odd (n = 2k + 1)
x^n + y^n = (x + y)(x^n-1 - x^n-2y +...- y^n-2x + y^n-1)
More Algebraic Formulas
2(a^2 + b^2) = (a + b)^2 + (a - b)^2
(a + b)^2 - (a - b)^2 = 4ab
(a - b)^2 = (a + b)^2 - 4ab
a^4 + b^4 = (a + b)(a - b)[(a + b)^2 - 2ab]
i hope it help you
Formulas involving squares
1) (a+b)²
=(a+b)(a+b)
=a²+ab+ab+b²
=a²+2ab+b²
Hence:
(a+b)²=a²+b²+2ab
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2) (a-b)²
=(a-b)(a-b)
=a²-ab-ab-b²
=a²-2ab+b²
Hence:
(a-b)²=a²+b²-2ab
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3) (a+b)²+(a-b)²
=a²+2ab+b²+a²-2ab+b²
=2a²+2b²
=2(a²+b²)
So : (a+b)²+(a-b)²=2(a²+b²)
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4) (a+b)²-(a-b)²
=a²+b²+2ab-(a²+b²-2ab)
=a²+b²+2ab-a²-b²+2ab
=4ab
(a+b)²-(a-b)²=4ab
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5) (a+b+c)²
=(a+b+c)(a+b+c)
=a(a+b+c)+b(a+b+c)+c(a+b+c)
=a²+ab+ac+ab+b²+bc+ac+bc+c²
=a²+b²+c²+2ab+2ac+2bc
=a²+b²+c²+2(ab+bc+ac)
a²+b²+c²+2(ab+bc+ac)
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6)(a+b)(a-b)
=a(a-b)+b(a-b)
=a²-ab+ab-b²
=a²-b²
(a+b)(a-b)=a²-b²
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7)(a-b-c)²
(a-b-c)(a-b-c)
=a(a-b-c)-b(a-b-c)-c(a-b-c)
=a²-ab-ac-ab+b²+bc-ac+bc+c²
=a²+b²+c²-2ab-2ac+2bc
=a²+b²+c²-2(ab-bc+ac)
(a-b-c)²=a²+b²+c²-2(ab-bc+ac)
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Hey I am a human being how can I prove every formula in a day:
Other such important formulas are as follows:-
(x+a)(x+b)=x²+x(a+b)+ab
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(x+a)(x-b)=x²+x(a-b)-ab
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(x-a)(x+b)=x²-x(a-b)-ab
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(x-a)(x-b)=x²-x(a+b)+ab
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a(x+a)(x+b)=x²+x(a+b)+ab
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a²+b²+c²-ab-bc-ac=1/2*[(a-b)²+(b-c)²+(c-a)²]
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Formulas involving cubes
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(a+b)³=a³+b³+3ab(a+b)
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(a-b)³=a³-b³-3ab(a-b)
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a³-b³=(a-b)(a²+b²+ab)
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a³+b³=(a+b)(a²-ab+b²)
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If (a+b+c)=0
then:
a³+b³+c³=3abc
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(a+b+c)(a²+b²+c²-ab-bc-ac)=a³+b³+c³-3abc
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(x+a)(x+b)(x+c)=x³+ax²+bx²+cx²+abx+bcx+acx+abc
==> (x+a)(x+b)(x+c)=x³+x(a+b+c)+(ab+bc+ac)
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(a+b+c)³=(a³+b³+c³)+3[(a+b+c)(ab+ac+bc)−abc]
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Other important formulas:
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For : ax²+bx+c
where x is the root of the equation
x satisfies the equation and makes the equation zero.
There may be more than one root in an equation.
Sum of roots of a quadratic equation is -b/a
Product of roots is c/a
For : ax³+bx²+cx+d
Let the roots be x,y and z
x+y+z=-b/a
xyz=-d/a
xy+yz+xz=c/a
For any polynomial
sum of roots=-b/a
product of roots=last term/first term
If the degree of polynomials are odd then the product is -last term/first term
If it is even then the product is last term/first term.
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Hope these formulas help you:-)
Please ask me if you have any doubts.