Algebra and Co-ordinate Geometry
In this chapter, we will learn about polynomials, linear equations in two variables and co-ordinate geometry.
Polynomials
Polynomials are those algebraic expressions in which the variables involved have only non-negative integral powers. In other words, a polynomial p(x) in one variable x is an algebraic expression in x of the form,
\[P(x)={{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+......+{{a}_{3}}{{x}^{3}}+{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}.\]
Where \[{{a}_{n}}\],\[{{a}_{n-1}}\],…..\[{{a}_{3}},{{a}_{2}},{{a}_{1}},{{a}_{0}}\]are \[{{a}_{n}}\]\[\ne \]0.
Here, \[{{a}_{n}}\],\[{{a}_{n-1}}\],….,\[{{a}_{3}},{{a}_{2}},{{a}_{1}},{{a}_{0}}\] are respectively the coefficients of \[{{x}^{n}},{{x}^{n-1}}\],….,\[{{x}^{3}},{{x}^{2}},x,{{x}^{0}}\] and n is called the degree of the polynomial.
Each of \[{{a}_{n}}{{x}^{n}},{{a}_{n-1}},{{x}^{n-1}}\],…..,\[{{a}_{3}}{{x}^{3}},{{a}_{2}}{{x}^{2}},ax,{{a}_{0}}\],is called a term of the polynomial p(x). The degree of the polynomial in one variable is the highest index of the variable in that polynomial.
Note:
(i) A non zero constant polynomial is a polynomial of degree 0. For example \[-3,\frac{2}{3,}\sqrt{5}\] etc are constant polynomials.
(ii) Constant polynomial 0 is called the zero polynomial. In such a polynomial all the constants are zero so degree of a zero polynomial is not defined.
(iii) For a polynomial p(x), a real number k is called a root (or zero) of the equation p(x) = 0 if p(k) =0.
Types of Polynomials
The following are the types of polynomials:
- Linear Polynomials (Polynomials of degree 1)
- Quadratic Polynomials (Polynomials of degree 2)
- Cubic Polynomials (Polynomials of degree 3)
- Biquadratic Polynomials (Polynomials of degree 4)
Remainder Theorem:
If p (x) is a polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial\[x-r\], then the remainder so obtained is p(x).
Factor Theorem:
For a polynomial p(x),
(i) If p(r) = 0 \[\Rightarrow \]( \[x-r\]) is a factor of p(x)
(ii) If (\[x-r\]) is a factor of p(x) \[\Rightarrow \] p(r) = 0
Algebraic Identities:
- \[{{(x+y)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy\]
- \[{{(x-y)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy\]
- \[{{x}^{2}}-{{y}^{2}}=(x+y)(x-y)\]
- \[(x+a)(x+b)={{x}^{2}}+(a+b)x=ab\]
- \[{{(x+y+z)}^{2}}={{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2xy+2yz+2zx\]
- \[{{(x+y)}^{3}}={{x}^{3}}+{{y}^{3}}+3xy(x+y)\]
- \[{{(x-y)}^{3}}={{x}^{3}}+{{y}^{3}}+3xy(x-y)\]
- \[{{x}^{3}}-{{y}^{3}}=(x-y)({{x}^{2}}+{{y}^{2}}+xy)\]
- \[{{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz=(x+y+z)\]
- \[({{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx)\]
Linear Equation in two variables
An equation is a statement in which one expression equals to another expression. An equation of the form \[ax+by+c=0\] where a, b and c are real numbers, such that a and b are both non-zero, is called a linear equation in two variables.
Solution of a Linear Equation in two Variables
A linear equation in two variables has infinitely many solutions.
The solution of a linear equation is not affected on
(i) Adding (or subtracting) the same number in both sides of the equation.
(ii) Multiplying (or dividing) the same non-zero number in both sides of the equation.
Graph of a Linear Equation in two Variables
General form of linear equation in two variables is \[ax+by+c=0\]
\[\Rightarrow by=-ax-c\] \[\Rightarrow y=\left( \frac{-a}{b} \right)x-\frac{c}{b}\]
The following steps are followed to draw a graph:
Step 1: Express x in terms of y or y in terms of x.
Step 2: Select at least three values of y or x and find the corresponding values of x or y respectively, which satisfies the given equation, write these values of x and y in the form of a table.
Step
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A line segment has a definite length.
Ray
A ray is a line segment extending indefinitely in one direction. A ray has no definite length. For example, in the shown figure \[\overrightarrow{PQ}\] is representing a ray having one and point P.
Line
A line is a ne is obtained on extending a line segment indefinitely in both the directions. In the shown figure, 0\[\overleftrightarrow{PQ}\] is represented as a line. A line has no end points so a line has no definite lengths.
Angle
An angle is generated when two rays originated from the same end point. In the shown figure POQ is the angle formed by two rays \[\]and \[\]. Here O is called the vertex of the \[\angle \]POQ and PO and OQ are called the arms of the angle POQ.
Note:
(i) If a ray stands on a line, then so formed adjacent angles are supplementary and its converse.
(ii) The vertically opposite angles formed by two intersecting lines are equal. more... 
Length of Arc and Area of a Sector
Let an arc AB makes an angle \[\theta <180{}^\circ \] at the center (O) of a circle of radius r, then we have:
