A number system is a structured method for representing and working with numbers. It encompasses a set of symbols, rules for arrangement and operations for performing arithmetic and mathematical manipulations. This chapter includes real numbers, their properties, polynomials, different types of polynomials and some important identities.
A number system is a method of representing or expressing numbers. The number system includes all these types of numbers.
Natural numbers are positive integers that begin at 1 and extend to infinity. They are the counting numbers used for basic counting. It does not include 0 or any negative numbers.
For example, 42 is a natural number.
Whole numbers include all non−negative integers, including zero and all positive counting numbers. They are the numbers used for counting objects and do not include fractions or negative values.
For example, 0, 14, 130 and 253 are all whole numbers.
An integer is a whole number, whether positive, negative, or zero, without any fractional or decimal parts.
For example, −1289, 0, 234 and 567 are integers, while 2.25, 0.258 and √5 are not integers.
Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. In other words, a rational number is of the form, a/b, where a and b are integers and b ≠ 0.
For example, 0.75 is a rational number as it can be expressed as 3/4.
Irrational numbers are real numbers which cannot be written as a fraction of two integers and their decimal expansions continue infinitely without repeating. In other words, they cannot be written in the form, a/b, where a and b are integers.
For example, √3, π, 2.02300789954…
Real numbers include both rational numbers, which include positive and negative integers as well as fractions and irrational numbers.
Properties of Irrational Numbers
If a and b are real numbers, then:
→ √ab = √a√b
→ (√a + √b)(√a − √b) = a − b
→ (a + √b)(a − √b) = a^{2} − b
→(√a + √b)(√c + √d) = √ac + √ad + √bc + √bd
→ (√a + √b)(√c − √d) = √ac − √ad + √bc − √bd
→ (√a + √b)^{2} = a + 2√ab + b
Laws of Exponents for Real Numbers
If a, b, m and n are real numbers, then:
→ a^{m} a^{n }= a^{m + n}
→ (am)^{n }= a^{mn}
→ a^{m}/a^{n }= a^{m − n}
→ a^{m}b^{m}=(ab)^{m}
Here a and b are the bases and m and n are exponents.
Exponential Representation of Irrational Numbers
Let x>0 be a real number and a and b be rational numbers, then
→ x^{a} × x^{b }= x^{a + b}
→ (x^{a})^{b }= x^{ab}
→ x^{a}/x^{b }= x^{a − b}→ x^{a}y^{a }= (xy)^{a}
Example: Solve 5^{√2 + 2}/25
a) 5
b) 5^{√2}
c) 25
d) 5^{√2}/5
Answer: b) 5^{√2}
Explanation: We know that x^{a} x^{b }= xa + b and 25 = 5^{2}
→ 5^{√2 + 2}/25
= (5^{√2} × 5^{2})/5^{2}
= 5^{√2}
A terminating decimal is a decimal number that ends after a finite number of digits following the decimal point. In other words, it comes to an end or terminates after a certain number of decimal places.
For example, 0.55, 9.124 and 3/8 are terminating decimals because they have a limited, definite number of digits after the decimal point.
A recurring decimal is a decimal representation of a fraction where one or more digits repeat indefinitely after a certain point. It is also called a repeating decimal.
For example, 5/6 is a recurring decimal.
A polynomial is a mathematical expression consisting of variables (indeterminates) and coefficients, combined using addition, subtraction and multiplication. They take the form a_{n}x^{n} + a_{n − 1}x^{n − 1} + a_{n − 2}x^{n − 2} + . . . + a_{1}x + a_{0}, where a_{0},a_{1},a_{2}, . . . ,a_{n − 2},a_{n − 1},a_{n} are coefficients, x is variable and n is a non−negative integer representing the degree of the variable.
For example, x^{2} + 2x + 1 is a polynomial.
→ A rational expression is an algebraic expression that is in the form of a fraction, where both the numerator and the denominator are polynomials. These expressions can be written as p(x)/q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0.
→ While every polynomial can be expressed as a rational expression, not every rational expression can be represented as a polynomial.
→ In a polynomial, the power of a variable must be a non−negative integer.
If p(x)=d(x)q(x) for some polynomial q(x), polynomial d(x) is called a divisor of a polynomial p(x).
The terms in a polynomial are the parts of the expression that are separated by addition or subtraction.
For example, if 12x^{2} + 6x + 4 is a polynomial, then 12x^{2},6x and 4 are the terms of the polynomial.
In a polynomial, a coefficient is simply the number that is multiplied by a variable.
For example, if 4x^{2} + 3x − 1 is a polynomial, then 4 is the coefficient of x2 and 3 is the coefficient of x.
→ A monomial is a polynomial expression composed of only one term. It can be a constant, a variable, or a product of constants and variables raised to non−negative integer power.
For example, 4, 6x, 2x^{3}
→ A binomial is a polynomial consisting of exactly two terms separated by addition or subtraction. It can be the sum or difference of two monomials. For example, 3x + 6, 4x − 3
→ A trinomial is a polynomial made up of three terms separated by addition or subtraction.
For example, 2x^{2} + 5x + 3
The degree of a polynomial is determined by the highest power of the variable within the expression.
For example, in 5x^{2} − 2x + 1, the degree is 2.
A linear polynomial is a polynomial of degree one. This means that it is an algebraic expression where the highest power of the variable is 1. It is also known as a first−degree polynomial.
For example, −5x + 2
A quadratic polynomial is a polynomial of degree two. This means that it is an algebraic expression where the highest power of the variable is 2. It is also known as a second−degree polynomial.
For example, 4x^{2} + 5x − 2
A cubic polynomial is a polynomial of degree three. This means that it is an algebraic expression where the highest power of the variable is 3. It is also known as a third−degree polynomial.
For example, 5x^{3} + 2x^{2} − x + 2
A biquadratic polynomial is a polynomial of degree four. This means that it is an algebraic expression where the highest power of the variable is 4.
For example, 2x^{4} + 3x^{3} − 6x^{2} + x − 2
Note
→ If substituting a real number 'a' into the polynomial p(x) results in p(a)=0, then 'a' is termed as a root of the equation p(x)=0.
→ Each linear polynomial in one variable has a distinct root.
→ The zero polynomial has every real number as its root.
→ A non−zero constant polynomial does not have any roots.
→ The concept of degree does not apply to the zero polynomial.
→ A non−zero constant polynomial has a degree zero.
The zeroes (or roots) of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if P(x) is a polynomial, then x=a is a zero of the polynomial if P(a)=0.
Example: If P(x)=x^{2} − 5x + 6 is a polynomial, then out of the following which is a root of the polynomial?
a) x = 1
b) x = -1
c) x = 2
d) x = -2
Answer: c) x = 2
Explanation: If P(x) is a polynomial, then x=a is a zero of the polynomial if P(a) = 0.
When x=1 the value of P(x) will be equal to
P(1) = (1)^{2} − 5(1) + 6
P(1) = 1 − 5 + 6
= 2 ≠ 0
Thus, x=1 is not a root of the polynomial.
Similarly, putting all options and checking the value of P(x)
When x=2 the value of P(x) will be equal to
P(2) = (2)^{2} − 5(2) + 6
P(2) = 4 − 10 + 6
= 0
Since P(x)=0 at x=2, we say that 2 is a zero of the polynomial x^{2} − 5x + 6.
The Factor Theorem provides a method for determining whether a given expression is a factor of a polynomial. It states that if f(x) is a polynomial and c is a constant, then x − c is a factor of f(x) if and only if f(c)=0. In simpler terms, if substituting c into the polynomial results in f(c)=0, then x − c is a factor of the polynomial.
For example, if f(x) = x^{2} − 5x + 4 we find that f(1) = 0, then x − 1 is a factor of f(x).
The Remainder Theorem states that if you divide a polynomial f(x) by x − c, where c is a constant, the remainder will be f(c). In simpler terms, if you substitute c into the polynomial, the result is the remainder.
If P(x) is a polynomial and P(x) is divided by x − c, then P(x) can be expressed as P(x)=(x − c)Q(x) + R(x), where Q(x) is the quotient and R(x) is the remainder.
Some Important Identities
→ (a + b)^{2} = a^{2} + 2ab + b^{2}
→ (a − b)^{2} = a^{2} − 2ab + b^{2}
→ (a + b)(a − b) = a^{2} − b^{2}
→ (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca
→ (a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}
→ (a − b)^{3} = a^{3} − 3a^{2}b + 3ab^{2} − b^{3}
→ a3 + b3 = (a + b)(a^{2} − ab + b^{2})
→ a^{3} − b^{3} = (a − b)(a^{2} + ab + b^{2})
→ a^{3} + b^{3} + c^{3} − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ca)
If a + b + c = 0 so, then a^{3} + b^{3} + c^{3}=3abc.
→ (x + a)(x + b) = x^{2} + (a + b)x + ab
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