The word polynomial is derived from the Greek words ‘poly’ which means ‘many‘ and ‘nominal’ which means ‘terms‘, so altogether it is said as “many terms”. A polynomial can have any number of terms but it is not infinite. In this chapter, we will deal with the basics of polynomials, their types and their significance in solving real-world problems.
A combination of constants and variables that are connected by some or all of these operations (+, −, × and ÷) is known as an algebraic expression.
Example: 7x^{2} + 8x + 4 is an algebraic expression containing three terms 7x^{2}, 8x and 4.
Terms - In an algebraic expression, each part is called a term.
Examples: 7x^{2}, 8x and 4 are terms in an algebraic expression 7x^{2} + 8x + 4.
Coefficients - Every term in an algebraic expression comes with a coefficient.
Examples: 7 is the coefficient of x^{2} and 8 is the coefficient of x in an algebraic expression 7x^{2} + 8x + 4.
Constants - A symbol having a fixed numerical value is called a constant.
Examples of Constants are 4, 9, ½, −3/5, 0.1, etc.
Variables - A symbol assigned with different unknown values is known as a variable.
Examples of Variables are x, y, z, a, b, x^{2}, y^{3}, etc.
A polynomial is an expression that is composed of variables, constants, coefficients, operators and exponents that are combined using mathematical operations such as addition, subtraction, multiplication and division.
The above polynomial 3x^{2}+ 5y − 9 includes:
(i) Monomial
A polynomial containing one term is called a monomial. [Mono means one.]
Examples of monomials are 5, 9, 7x, 3x^{2}, 8x^{3}, 9x^{4}, etc.
(ii) Binomial
A polynomial containing two non-zero terms is called a binomial. [Bi means two.]
Examples of binomials are (7 + 3x), (8x + 6x^{2}), (2 + x^{3}), (x^{4} + 3), etc.
(iii) Trinomial
A polynomial containing three non-zero terms is called a trinomial. [Tri means three.]
Examples of trinomials are (3x^{3} + 8x^{2} + 4), (9x^{4} + 2x^{2} + 6), (3x^{6} + 7x^{4} + x), etc.
The degree of a polynomial is the highest exponent of a monomial within a polynomial. Thus, a polynomial having one variable that has the largest exponent is called a degree of the polynomial.
For an expression to be a polynomial, the exponent of the variable must be a non-negative integer.
An algebraic expression of the form p(x) = a_{0}+ a_{1}x + a_{2}x^{2} +…………+ a_{n−1}x^{n−1} + a_{n}x^{n} is called a polynomial in one variable (x) of degree n.
Where a_{0}, a_{1}, a_{2},…, a_{n−1}, a_{n} are real numbers, a_{n} ≠ 0 and n is a non-negative integer,
Here, a_{0} is called the constant term of the given polynomial.
a_{1}, a_{2},,…, a_{n−1}, a_{n} are called the coefficients of x, x^{2},........, x^{n−1} and x^{n}, respectively.
Rational Expression: A rational expression is in the form p(x)/q(x) where both p(x) and q(x) are polynomials and q(x) is not equal to zero.
Relationship between Polynomials and Rational Expressions: Every polynomial is a type of rational expression but not every rational expression is a polynomial. Rational expressions can have more complex forms than simple polynomials.
Power of Variables in Polynomials: In a polynomial, the powers of variables must be whole numbers. This means the exponents (powers) attached to the variables are integers.
Divisors in Polynomials: If a polynomial d(x) can be multiplied by another polynomial q(x) to get a third polynomial p(x), then d(x) is called a divisor of p(x). This is represented as p(x) = d(x)q(x).
A zero polynomial is a polynomial consisting of one term, namely zero. The degree of a zero polynomial is not defined.
Examples of zero polynomials are 0x^{3} + 0x^{2} + 0x + 0, 0x^{2} + 0x + 0, 0x + 0, 0, etc.
A constant polynomial is a polynomial containing one term consisting of a non-zero constant. Every non-zero real number is a constant polynomial. A polynomial of degree 0 is a constant polynomial.
Examples of constant polynomials are 7, 9x^{0}, −4, ^{5}⁄6, 2.05, π, etc.
A polynomial of degree 1 is called a linear polynomial.
A linear polynomial in x is of the form (ax + b), where a and b are real numbers and a ≠ 0.
Examples:
(i) 3x + 8 is a linear polynomial in variable x.
(ii) 7y − 6 is a linear polynomial in variable y.
(iii) 3x + 7y − 6 is a linear polynomial in two variables x and y.
A polynomial of degree 2 is called a quadratic polynomial.
A quadratic polynomial in x is of the form ax2 + bx + c, where a, b, c are real numbers and a ≠ 0.
Example: 7x^{2} + 5x + 6 is a quadratic polynomial in variable x.
A polynomial of degree 3 is called a cubic polynomial.
A cubic polynomial in x is of the form ax^{3} + bx^{2} + cx + d, where a, b, c, d are real numbers and a ≠ 0.
Example: 7x^{3} + x^{2} + 6x + 2 is a cubic polynomial in variable x.
A polynomial of degree 4 is called a biquadratic polynomial.
A biquadratic polynomial in x is of the form ax^{3} + bx^{2} + cx + d, where a, b, c, d are real numbers and a ≠ 0.
Example: 3x^{4} − 7x^{3} + x^{2} + 6x + 2 is a biquadratic polynomial in variable x.
A polynomial is in standard form when it is written with the powers of x either in ascending or descending order.
Examples: x + 1, x^{2} − x + 1, 2x^{3} + x^{2} − 3x + 1, 1 + 2x − x^{2} + x^{3} + x^{4}, etc.
Important to know:
Is 1+^{1}⁄x a polynomial?
1+^{1}⁄x may be written as x + x^{−1}. Here, the exponent in one term of x is –1 which is a negative integer. Thus, 1+^{1}⁄x is not a polynomial.
The value of a polynomial p(x) at x = a is obtained by putting x = a in p(x) and it is denoted by p(a).
Example: If p(x) = 4x^{2} + 3x + 2, what is the value of p(−2)?
a) 6
b) 12
c) 18
d) 24
Answer: b) 12
Explanation: If p(x) = 4x^{2} + 3x + 2, then
p(−2) = 4 × (−2)^{2} + 3 × (−2) + 2
= 16 − 6 + 2
= 12
If p(x) is a polynomial, then p(a) = 0.
Here, ‘a’ is a root of the polynomial p(x). It is also known as the zeros of the polynomial p(x).
Finding the zeros of a polynomial p(x) means solving the equation p(x) = 0.
Are 3 and −3 the zeros of the polynomial (x − 3)?
Polynomial p(x) = (x − 3)
Put the value of x in the polynomial.
Therefore, p(3) = 3 − 3 = 0
and p(−3) = −3 − 3 = −6 ≠ 0
∴ 3 is a zero of (x − 3) and −3 is not a zero of (x − 3).
Note:
→ Every linear polynomial in one variable has a unique root (zero).
→ A non-zero constant polynomial has no root (zero).
→ Every real number is a root (zero) of the zero polynomial.
p(x) and g(x) are two polynomials such that degree of p(x) ≥ degree of g(x) and g(x) ≠ 0.
As we know, Dividend = (Divisor × Quotient) + Remainder
On dividing p(x) by g(x), q(x) is the quotient and r(x) is the remainder.
p(x) = g(x) × q(x) + r(x)
where r(x) = 0 or degree of r(x) < degree of g(x)
If p(x) is divided by (x − a), then the remainder is p(a).
Here, p(x) is a polynomial of degree 1 or more and ‘a’ is any real number.
Note: If p(x) is divided by (x + a), then the remainder is p(−a).
Example: What is the remainder when the polynomial p(x) = 2x^{4} + 3x^{3} − 4x^{2} + x − 1 is divided by g(x) = x − 3?
a) 199
b) 209
c) 219
d) 229
Answer: b) 209
Explanation: By the remainder theorem,
we know that if p(x) is divided by (x − a), then the remainder is p(a).
Given: p(x) = 2x^{4} + 3x^{3} − 4x^{2} + x − 1
g(x) = x − 3
If (2x^{4} + 3x^{3} − 4x^{2} + x − 1) is divided by (x − 3), the remainder will be p(3).
Thus, to get the remainder of the expression, we need to put x = 3 i.e. p(x) = p(3).
∴ p(x) = p(3) = 2 × 3^{4} + 3 × 3^{3} − 4 × 3^{2} + 3 − 1
= 162 + 81 − 36 + 3 − 1
= 209
∴ The required remainder is 209.
If p(x) is a polynomial of degree 1 or more and a is any real number, then
(i) When p(a) = 0, (x − a) is a factor of p(x).
(ii) When (x − a) is a factor of p(x), p(a) = 0.
Note:
→ If x − 1 is a factor of a polynomial of degree 'n' then the sum of its coefficients is zero.
If p(x) = x^{3} − 2x^{2} − 4x + 5 and g(x) = x − 1, then check whether g(x) is a factor of p(x) or not.
We write g(x) = 0 to get the value of x for which p(x) must be zero if g(x) needs to be a factor of p(x).
∴ g(x) = 0
⇒ x − 1 = 0
∴ x = 1
By the factor theorem, if g(x) = (x − 1) to be a factor of p(x), then p(1) = 0.
p(x) = x^{3} − 2x^{2} − 4x + 5
∴ p(1) = 1^{3} − 2 × 1^{2} − 4 × 1 + 5
= 1 − 2 − 4 + 5
= 6 − 6
= 0
Hence, if p(1) = 0, then g(x) is a factor of p(x).
Factorisation is the process of writing an algebraic expression as the product of two or more algebraic expressions. Important identities which help in factorisation are as follows:
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