POLYNOMIALS
Introduction
1. "Term" : A Mathematical symbol consisting of a sign part, Numerical
(Number) Part and a variable part is called a "Term"
TERM NAME Sign Part Numerical Part Variable
Part Degree Coefficient
2 Constant
Term 2 x0 0 +2
-2x x- term 2 x 1 -2
-5/6 x2 x2 – term 5/6 x2 2 -5/6
1/3 x3 x3- term 1/3 x3 3 +1/3
Example : 2x, -3, ½x²,x³.etc
Note
* Terms are named after (based on) their variable parts.
* Every term has a co-efficient, base and degree (index, power, exponant,
radical).
* The Co-efficient is the combination of sign part and the numberical part of a
term.
* Sentence: A group of terms is called a sentence.
Ex: x+2, x²+7x+2
* Open sentence: Any sentence which consists of at least one variable in it
Is called an open sentence.
Ex: x+1=0, x²+2x+3
Note:
* Expressions and Equations are the two types of open sentences.
* Expression : An open sentence which doesn't consists of equality sign (=) is
called an expression.
Ex: x+2, x²-3, x³-x²+3x+1.
* Equation : An open sentence which consists of equality sign (=) is called
an equation.
Ex: x+2=0, x²-3=0
* Polynomial : Any nth degree expression is called a polynomial
Note:
(i) Any polynomial which is written in the decending orders of x is called
the standard form or the general form. of a polynomial and it is denotes by
f(x), p(x), g(x), h(x) ........ etc.
i.e f(x)= aₙxⁿ + aₙ₋₁xⁿ⁻¹ + aₙ₋₂x² + ... ... + a₀x⁰ where
aₙ, aₙ₋₁, aₙ₋₂,........a₀ are called the real coffecients of xⁿ, xⁿ⁻¹ .........respectively.
Types of Polynomials
Based on highest degree
* Linear Polynomial: Any polynomial which is in the form of ax + b, where
a ≠ 0 and a, b being real numbers is called a linear
polynomial.
Ex : 2x + 3, 5x + 4, -5x - 3
Note :
* 'ax + b' is also called the standard form or general form of a linear
polynomial.
* Every linear polynomial has two terms.
* The first term 'ax' is called the x-term.
* The second term b is called the xº term (or) constant term (or)
independent term.
* The highest degree of every linear polynomial is '1' (one).
* Every linear polynomial has only one solution called factor (itself).
* Quadratic Polynomial : Any polynomial which is in the form of ax² + bx + c
where a ≠ 0 and a, b, c being real numbers is called a
quadratic polynomial.
Example: x² + 3x + 2, 2x² - 5x - 7, x² - 5x + 6 etc.
Note :
* ax² + bx + c is called the standard form (general form) of a quadratic
polynomial.
* Every quadratic polynomial has three terms in it.
* The first term ax² is called 'x²' term.
* The second term bx is called 'x' term.
* The third term c is called xº term or constant term or independent term.
* Every quadratic polynomial has exactly two solutions called factors
* If the two zeros of a quadratic polynomial f(x) are denoted by α, β, then the
formula to find it is given by
f(x)= k [x²- (sum of the zeros) x + (product of the zeros)]; i.e.
f(x)= k [x² - (α + β) x + αβ]; where the relationship between the zeros and
the co-efficients is given by
α + β = -(co-efficient of x )/(co-efficient of x^2 )= -b/a and αβ = (constant term )/(co-efficient of x²)= + c/a
* Cubic polynomial: Any polynomial which is in the form of ax³ + bx² + cx +
d, where a ≠ 0 and a, b, c, d being real numbers is called a
cubic polynomial.
Ex: x³ - 3x² - 3x + 1, x³ + 5x² - 2x + 10
Note :
* ax³ + bx² + cx + d is also called the standard form of a cubic polynomial.
* Every cubic polynomial consists of 4 terms.
* The first term ax³ is called x³ term.
* The second term bx² is called x² term.
* The third term cx is called x term.
* The fourth term d is called xº term or constant term or independent term.
* Every cubic polynomial has exactly 3 solutions called factors.
* If the three zeros of a cubic polynomial f(x) are denoted by α, β, γ, then the
formula to find it is given by
f(x)= k [x³- (sum of the zeros)x² + (sum of product of zeros taken two at a
time)x - (product of the zeros)]; i.e.
f(x) = k [x³- (α + β + γ)x² + (αβ + βγ + γα) x - αβγ]; where the
relationship between the zeros and the co-efficients is given by:
α + β + γ= -(co-efficient of x^2 )/(co-efficient of x^3 )= -b/a and αβγ = -(constant term)/(co-efficient of x^3 )= -d/a
αβ + βγ + γα = -(co-efficient of x )/(co-efficient of x^3 )= c/a
* Biquadratic Polynomial: Any polynomial which is in the form of ax⁴+ hx³+
cx² + dx + e where a, b, c, d, e, being Real numbers,
a ≠ 0 is called a biquadratic polynomial.
Ex: 6x⁴+ 3x³-2x²+6x+1
Note:
• The standard form of a biquadratic polynomial is ax⁴+ hx³+ cx²+ dx + e.
• Every biquadratic polynomial has 5 terms.
• The first term ax⁴ is called 'x⁴' term.
• The second term bx³ is called 'x³' term.
• The third term cx² is called 'x²' term.
• The fourth term dx is called 'x' term.
• The fifth term e is called 'x⁰' term or constant or independent term.
• Every biquadratic polynomial has exactly 4 solutions called factors.
Types of Polynomials
Based on number of terms
• Monomial: Any polynomial which has 1 term is called a monomial.
Ex: x², 2xy, -x, xyz.......etc.
• Bionomial: Any polynomial which has 2 terms is called a Bionomial
Ex: x + 2, 3x² + 5, 4x² + 5, 4x³ + 2x² ........etc.
• Trinomial: Any polynomial which has 3 terms is called a trinomial
Ex: x² + 3x + 2, x + y + z, ...... etc.
• Zero of a polynomial: If f(x) be any given polynomial and 'a' being a real
number and if f(a)=0 then a is called the zero of the
polynomial f(x).
• Factorisation: A procedure (method) in which a given polynomial is
expressed as the product of its maximum number of factors is
known as factorisation.
Solve : Finding all the possible values of the unknown quantity or
quantities, which exists in the problem is called solving.
Remainder Theorem: If f(x) be any given polynomial of degree greater
than or equal to 1 and small 'a' be any real number,
such that f(x) is divided by (x - a), then the
remainder is equal to f(x).
f(x) = (x - a) q(x) + r(x)
Note:
* f(x) is called the dividend
* (x - a) is called the divisor
* q(x) is called the quotient
* r(x) is called the remainder.
* If a polynomial p(x) is divided by (x + a) the remainder is the value of p(x)
at x = -a i.e., p(-a).
[x + a = 0 => x = -a].
* If a polynomial p(x) is divided by ax - b the remainder is the value of p(x) at
x = -a i.e., p(-a).
[x + a = 0 => x = -a]
* If a polynomial p(x) is divided by ax + b the remainder is the value of p(x)
at x = -b/a i.e., p(-b/a).
* If a polynomial p(x) is divided by (b - ax) the remainder is the value of p(x)
at x = b/a i.e., p(-b/a).
[b - ax = 0 => -ax = -b => x = -b/-a => x = b/a]
* Remainder theorem is used to find the remainder.
* Remainder theorem is also called the division rule i.e.
Dividend = Division x Quotient + Remainder.
* Factor Theorem : If f(x) be any given polynomial of degree greater than or
equal to 1 and small 'a' be any real number, such that f(a)
= 0, then (x - a) is a factor of f(x).
Note:
* (x + a) is a factor of a polynomial f(x) if f(-a) = 0
* (ax - b) is a factor of a polynomial f(x) if f(b/a) = 0
* (ax + b) is a factor of a polynomial f(x) if f(-b/a) = 0.
* (x - a) is a factor of a polynomial f(x) if f(a) = 0
* Factor theorem is used to factorize a polynomial by division method or
Horner method.
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