Algebraic Expression for SBI PO
Algebraic Expression for SBI PO
Algebraic Expression : Algebraic expression for SBI PO is an expression that contains variables , constants and algebraic operations (addition, subtraction, multiplication, division)
Example x2+ 3y = 7
2xy = 61
Polynomial: An algebraic expression for SBI PO which can be written in the form
p(x) = a0+ a1 x + a2 x2 + ………………+ an xn is called a polynomial in x, where a0,a1, ….., an belongs to real numbers and n is a non – negative integer.
I.e. an algebraic expression which contains only non -negative powers of x is called a polynomial in x.
Example: 2x3 – 4x + 5
x4 + 2x2 – x + 2
Theory for Algebraic Expression for SBI PO
Degree of a polynomial: The highest power of x present in a polynomial is called the degree of a polynomial.
Example: 2x3 – 4x + 5 _The highest power of x in this polynomial is 3, so its degree is 3
x4+ 2x2– x + 2 _ the highest power of x in this polynomial is 4, so its degree is 4
Constant polynomial: A polynomial of degree 0 is called a constant polynomial
Example: 4
56
-8/5
Zero polynomial: p(x) = 0 is called a zero polynomial. The degree of zero polynomial is not defined
Linear polynomial: A polynomial of degree 1 is called a linear polynomial.
Example: x + 3
2x – 5
Quadratic polynomial: A polynomial of degree 2 is called a quadratic polynomial.
Example: 4x2 + x + 6
x2 – 4
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Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial.
Example: x3 – 4x2 + x + 6
10x3+ 85
Remainder theorem: Let p(x) be any polynomial and let a be any real number . if p(x) is divided by the linear polynomial x – a , then the remainder is p(a).
In general when p(x) is divided by the linear polynomial ax + b , the remainder is p(-b/a).
Example: if we divide, p(x) = x3– 4 x2+ x + 6 by 2x + 4 the remainder is p(-4/2) = p(-2)
p(-2) = (-2)3– 4(-22) + (-2) + 6
= – 8 – 16 – 2 + 6
= -20
Factor theorem: If p(x) is a polynomial of degree n ≥ 1 and a is any real number, then x – a is a factor of p(x), if (if and only if ) p(a) = 0
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Example: Check whether x + 2 is a factor of p(x) = x2 + 4x + 4
We first put x + 2 = 0
x = -2
Now we calculate p(-2)
p(-2) = (-2)2+ 4(-2) + 4
= 4 – 8 + 4 = 0
p(-2) = 0
⇒ x + 2 is a factor of x2+ 4x + 4
Greatest integer: The greatest integer of a real number x is the greatest integer less than or equal to the x. It is denoted by x
Example: [4.72] = 4
[62] = 62
[-1.23] = -2
Minimum value of a quadratic polynomial: Minimum value of a quadratic polynomial ax2 + bx + c is given by the formula
4ac – b2/4a
Algebraic Expression for SBI PO
- (a + b )2 = a2 + b2 + 2ab
- (a – b )2 = a2 + b2 – 2ab
- a2 – b2 = (a + b) (a – b)
- (a + b )3 = a3 + b3 + 3ab(a + b)
- (a – b )3 = a3 – b3 – 3ab(a – b)
- a3 + b3 = (a + b) ( a2 + b2 – ab)
- a3 – b3 = (a – b) ( a2 + b2 + ab)
- a3 + b3 + c3 – 3abc = (a + b + c) (a2+ b2 + c2– ab – bc – ca)
- a3 + b3 + c3 – 3abc = 1/2 (a + b + c) [(a – b )2 + (b – c )2+ (c – a )2]
- (a + b + c) (a2+ b2 + c2– ab – bc – ca) = 1/2 (a + b + c) [(a – b )2 + (b – c )2+ (c – a )2]
- If a + b + c = 0, then a3 + b3 + c3= 3abc
- (a + b + c)3 = a3 + b3+ c3+ 3(a + b) (b + c) (c + a)
- (a + b + c)2= a2 + b2+ c2 + 2ab + 2bc + 2ca
Factors of some polynomial
- xn + yn is exactly divisible by (x + y) only when n is odd.
Example x5 + y5 is exactly divisible by x + y
- xn + yn is not exactly divisible by x + y when n is even
- xn + yn is never divisible by x – y
- xn – yn is exactly divisible by x + y when n is even
- xn + yn is exactly divisible by x – y
Quadratic equations : A general form of quadratic equation is ax2 + bx + c = 0, where a,b and c are real numbers and a 0
Example 3x2 + 5x – 8 = 0
Roots or zeros of quadratic equations : The values of x which satisfies the given equation are called its roots.
Example x = 4 and x = -3 are the roots of the equation x2 – x – 12 = 0
A quadratic equation has exactly two roots
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Methods of finding the roots of quadratic equations
- Method of splitting the middle term suppose we have to find the roots of the equation x2 – x – 12 = 0
Find the product of constant term and coefficient of x2
-12*1 = -12
Now find two numbers whose product is -12 and sum is -1 (coefficient of x)
That is -4 and 3
x2 – x – 12 = x2 – 4x + 3x – 12 = 0
= x(x – 4) + 3(x – 4) = 0
= (x – 4) (x + 3) = 0
= x – 4 = 0 or x + 3 = 0
= x = 4 or x = -3
- By quadratic formula or sridharacharya formula : The quadratic equation ax2 + bx + c = 0 where a,b and c R and a 0 has two roots
X = -b + b2 – 4a c2 a and x = -b – b2 – 4a c2 a
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Discriminant of a quadratic equations : b2 – 4ac is called the discriminant of the quadratic equation ax2 + bx + c = 0 where a,b and c R and a 0 and is denoted by D.
Nature of roots of the quadratic equation :
- When D < 0, both roots will be imaginary
- When D = 0, both roots will be real and equal
- When D > 0, both roots will be real and distinct
Consider a quadratic equation
Y = x2 + 2x – 1
Graph of this equation is given below
The graph cut the x axis between (-2 and -3) and between (0 and 1)
So the roots of this equation lies exactly where graph cut the x axis
-1 + 2 and -1 – 2
Consider the equation y = x2 – 5x + 6
The roots of this equation are 2 and 3
Because the graph cut the x axis at 2 and 3
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