Sample Material of Online Coaching For SSC CGL (Tier - 2) - Basic Algebra
Sample Material SSC CGL TIER-2 Online Coaching
Numerical Aptitude (Chapter: Basic Algebra)
Polynomials
An expression in term of some variable(s) is called a polynomial.
For example
f(x) = 2x – 5 is a polynomial in variable x
g(y) = 5y2 – 3y + 4 is a polynomial in variable y
Note that the expressions like 3x2-Öx, 1 ,5x3- 4 etc. are not polynomials. x2-7x+6 x
Thus, a rational x integral function of ‘x’ is said to be a
polynomial, if the powers of ‘x’ in the terms of the polynomial are neither
fractions nor negative. Thus, an expression of the form
f(x) = an xn + an–1xn–1 + … + alx
+ a0 is called a polynomial in variable x where n be a positive
integer and a0, al, ...,an be constants (real
numbers).
Degree of a Polynomial
The exponent of the highest degree term in a polynomial is known as its degree.
For example
f(x) = is a polynomial in the variable x of degree 1.
p(u) = 3u3 + u2 + 5 – 6 is a polynomial in the variable u
of degree 3.
q(t) = 5 is a polynomial of degree zero and is called a constant polynomial.
Linear Polynomial
A polynomial of degree one is called a linear polynomials. In general f(x) = ax + b, where a ¹ 0 is a linear polynomial.
For example
f(x) = 3x – 7 is a binomial as it contains two terms.
Quadratic Polynomials
A polynomial of degree two is called a quadratic polynomials. In general f(x) = ax2 + bx + c, where a ¹ 0 is a quadratic polynomial.
For example
f(x) = x2 – 7x + 8 is a trinomial as it contains 3 terms
Cubic Polynomial
A polynomial of degree 3 is called a cubic polynomial in general.
f(x) = ax3 + bx2 + cx + d, a ¹
0 is a cubic polynomial.
For example
f(x) = 2x3 – x2 + 8x + 4
Biquadratic Polynomial
A fourth degree polynomial is called a biquadratic polynomial in general.
f(x) = ax4 + bx3 + cx2 + dx + e, a
¹ 0 is a biquadratic polynomial.
Zero of a Polynomial
A real number a is a zero (or root) of a polynomial f(x), if f (a) = 0
For example
If x = 1 is a root of the polynomial 3x3 – 2x2 + x –
2, then f(l)= 0
f(x) = 3x3 – 2x2 + x – 2, f(1) = 3 × 13 – 2 × 12 + 1 – 2 =
3 – 2 + 1 – 2 = 0,
As f(1) = 0
x = 1 is a root of polynomial f(x)
(1) A polynomial of degree n has n roots.
(2) A linear polynomial of f(x) = ax + b, a ¹ 0 has a
unique root given by x =
(3) Every real number is a root of the zero polynomial.
(4) A non-zero constant polynomial has no root.
Remainder Theorem
Let f(x) be a polynomial of a degree greater than or equal to one and a be any real number, if f(x) is divisible by (x – a), then the remainder is equal to f(a) .
Example 1: Find the remainder when f(x) = 2x3 – 13x2
+ 17x + 10 is divided by x – 2.
Solution. When f(x) is divided by x – 2, then remainder is given by
f(2) = 2(2)3 – 13(2)2 + 17(2) + 10 = 16 – 52 + 34 + 10 = 8
Thus, on dividing f(x) = x3 – 13x2 + 17x + 10 by x – 2, we
get the remainder 8.
Factor Theorem
Let f(x) be a polynomial of degree greater than or equal to
one and a be any real number such that f(a) = 0, then (x – a) is a factor of f(x).
Conversely, if (x – a) is a factor of f(x), then f(a) = 0.
Example 2: Show that x + 2 is a factor of the polynomial x2
+ 4x + 4.
Solution. Let f(x) = x2 + 4x + 4 = (x + 2)2 = {x –
(–2)} is a factor of f(x) if f(–2) = 0
Now, f(–2) = (–2)2 + 4(–2) + 4 = 4 – 8 + 4 = 0
Hence, x + 2, is a factor of f(x).
:: Home Assignment for Practice ::
1. Which of the following expression is a polynomial in one variable?
(a) 3x2 –2y + 3
(b) –x2 + 6x – 5 = 0
(c) x2 + y2 = –3
(d) 2x2 – 2y -1 = 0
2. Which of the following expression is a polynomial in two variables?
(a) x2 – 2xy + y2 – x + y
(b) x3 + y3 + z3 –1
(c) x2 + 3x + 2
(d) x2 – 2x 2 + 3x + 5
3. The degree of the polynomial 2x3 – 5x2 + x – 3 is
(a) 2
(b) 3
(c) 1
(d) 0
4. The degree of the polynomial is
(a) 1
(b) 2
(c) 0
(d) 4
5. The degree of the polynomial 7 is
(a) 1
(b) 0
(c) 3
(d) 2