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| Some Basic Definitions |
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| An expression of the form |
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| in which ai are numbers belonging to some number system and 'n' is a non-negative integer, is called a polynomial. ai are called the
coefficients of if ao ≠ 0,
the polynomial is said to be of degree n. |
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| If x is replaced by a number from a number system to which ai belong, we get a number called a value of a polynomial. |
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| Usually a polynomial is denoted by P(x) and if k is any number then P(k) denotes the value of P(x) at x=k. Then a polynomial can be used to define a function with x or y or any letter of the alphabet. The coefficient of the polynomial may belong to any system of numbers. |
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| If ao
≠ 0, then the polynomial P(x) or f(x) is
of degree 'n' i.e., it is the highest power of the variable x in the rational and integral polynomial. A polynomial of degree 2 is called a quadratic polynomial. |
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| An equation in a single variable is called a polynomial equation of degree 'n' if it is an equation of the form |
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| In other words, an equation is a statement of equality between two algebraic expressions. It is satisfied by a limited number of values of the variable. |
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| Example: |
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| i) 3x + 5 = 8, is satisfied by the value of x = 1. |
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| ii) x2 - 4 = 0, is satisfied by the value of x = 2 or x = -2. |
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| An algebraic identity is a statement of equality between two algebraic expressions, but it is satisfied for all values of the variable. |
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| Example: |
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is satisfied for all the values of x. The sign  is used to distinguish an identity from an equation. |
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| An equation of the type |
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| where a, b, c are constants is called a quadratic equation in the variable x or an equation of the second degree. |
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| In the above equation, |
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| a is the coefficient of x2 |
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| b is the coefficient of x |
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| and c is the constant term or absolute term. |
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| Examples: |
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| are called Pure quadratic equation. |
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| are called adfected quadratic equations. |
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| are called adfected quadratic equations. |
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| ax2 + bx + c = 0 is called a general quadratic equation, a, b, c are |
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| There are four methods of solving quadratic equations. |
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| i) By factorization |
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| ii) By completing the squares |
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| iii) By using the formula |
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| iv) By graphing |
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| A root of the equation f(x) = 0 is that value or values of x which make f(x) = 0. In other words, x = a or x = b are said to be the root of f(x) = 0, if f(a) = 0, and f(b) = 0 |
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| i.e., in f(x) = 0, replace x either by a or by b. |
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| The determination of all the roots of a given equation is called the solution of the equation. |
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