CBSE Class 9

On this page, you will find Coordinate Geometry Class 9 Notes Maths Chapter 6 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 6 Coordinate Geometry will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 6 Notes Coordinate Geometry

Coordinate Geometry Class 9 Notes Understanding the Lesson

Rene Descartes was a French mathematician. He introduced an idea of Carterian Coordinate System for describing the position of a point in a plane. The idea which has given rise to an important branch of Mathematics known as Coordinate Geometry.

1. Cartesian coordinate system: A system which describe the position of a point in a plane is called Cartesian system.

2. Cartesian coordinate axis: Let us draw a horizontal line XX’ and a vertical line YY’ in a plane. Both the lines intersect each other at 90°, then the plane is divided into four parts.

The lines XX’ and YY’ are called axes i.e., XX’ is the x-axis and YY’ is y-axis.

3. Origin: The point where both the axis intersect each other is known as origin.

4. Quadrant
When XX’ and YY’ intersect each other then the plane is divided into four parts. These parts are called quadrants. The plane is known as Cartesian plane or XY plane.

5. Coordinate Geometry: It is a branch of geometry in which geometric problems are solved through algebra by using coordinate system.

6. Cartesian Coordinate (Rectangular Coordinate) System

In this system, the position of a point P is determined by knowing the distances from two perpendicular lines passing through the fixed point O is called origin.
The position of the point P from origin on x-axis is called x-coordinate and the position of P from origin on y-axis is called y-coordinate.

Abscissa: The distance of a point P from y-axis is called abscissa.

Ordinate: The distance of a point P from x-axis is called its ordinate.

Abscissa and ordinate together determine the position of a point in a plane, and it is called coordinates of the point. If a and b are respectively abscissa and ordinate, then the coordinates are (a, b).

Note:

• In first quadrant values of x and y are both positive.
• In second quadrant value of x is negative whereas the value of y is positive.
• In third quadrant value of x and y both are negative.
• In fourth quadrant, the value of x is positive and value ofy is negative.
• Perpendicular distance of a point from x-axis = (+)y-coordinate.
• Perpendicular distance of a point from y-axis = (+)x-coordinate.
• A point which lies on x-axis has coordinates of the form (a, 0).
• A point which lies on y-axis has coordinates of the form (0, b).
• Distance of a point P(x, y) from origin 0(0, 0) =\(\sqrt{x^{2}+y^{2}}\)
  e.g., distance of a point A(4,5) from origin, OA = \(\sqrt{4^{2}+5^{2}}\)
  \(=\sqrt{16+25}=\sqrt{41}\)units

On this page, you will find Triangles Class 9 Notes Maths Chapter 5 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 5 Lines and Angles will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 5 Notes Triangles

Triangles Class 9 Notes Understanding the Lesson

Two geometric figures are said to be congruent if they have exactly the same shape and size.
Note: Congruent means equal in all respect. When one figure is kept over another then it should superimpose on the other to cover it exactly.

If a 500-rupee note is placed over another 500-rupee note then they cover each other.
If 5-rupee coin is placed over another 5-rupee coin of same year, then they cover each other completely.
Congruence of line segments: Two line segments are congruent if they are of the same length. Length of AB = length of CD

Hence, \(\overline{\mathrm{AB}} \cong \overline{\mathrm{CD}}\)

Congruence of angles: Two angles are congruent if they have equal degree measures.

Hence, \(\angle \mathrm{ABC} \cong \angle \mathrm{CDE}\)

Congruence of squares: Two squares are said to be congruent, if they have equal sides.
Hence,

Note: Congruent plane figures are equal in area.

Congruence of circles: Two circles are congruent if they have equal radii.
Hence, Circle C₁ ≅ Circle C₂

Congruent Polygons
Two polygons are said to be congruent if they are the same size and shape. For existence of congruency,
(a) their corresponding angles are equal, and
(b) their corresponding sides are equal.

Congruence Triangles
Two triangles are congruent if they will have exactly the same three sides and three angles.

Axiom 7,1: SAS (Side-Angle-Side) Congruence rule: Two triangles are said to be congruent if two sides and the included angle of one triangle are equal to A D
the two sides and the included angle of the other.

In ΔABC and ΔDEF,
AB = DE
∠ABC = ∠DEF
BC = EF
ΔABC ≅ ΔDEF (by SAS)

Theorem 7.1: ASA (Angle-Side-Angle)
Congruence rule: Two triangles are said to be congruent, if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.
In Δ ABC and Δ DEF,
∠ABC = ∠DEF
BC = EF
∠ACB = ∠DFE
∴ΔABC = ΔDEF (by ASA)

Given: AABC and ADEF in which
∠ABC = ∠DEF, ∠ACB = ∠DFE and BC = EF
To Prove: ΔABC = ΔDEF
Proof: There are three cases arises for primary two congruence of the two triangles.

Case I: Let AB = DE
In ΔABC and ΔDEF,
AB = DE (assumed)
∠ABC = ∠DEF (given)
BC = EF (given)
So ΔABC ≅ ΔDEF (by SAS congruency)

Case II: Let AB > DE. So we can take a point P on AB such that PB = DE.
Now in ΔPBC and ΔDEF
PB = DE (by construction)
∠PBC = ∠DEF (given)
BC = EF (given)
∴ ΔPBC ≅ ΔDEF (by SAS Congruency)
∠PCB = ∠DFE …(1) (by CPCT)
∴ ∠ACB = ∠DFE …(2)
From eqn (1) and (2),
∠PCB = ∠ACB
which is not possible. This is only possible if point P coincides with A.
Hence AB = DE (PB=AB)
So ΔABC = ΔDEF (by SAS congruency)

Case III: If AB < DE, so we take a point Q on DE such that QE = AB
Now in ΔABC and ΔQEF,
AB = QE (by construction)
∠ABC = ∠QEF (given)
BC = EF (given)
Hence ΔABC = ΔQEF (by SAS congruency)
∠ACB = ∠QFE (by CPCT)
But ∠ACB = ∠DFE
Hence ∠QFE = ∠DFE
which is only possible if point Q coincides with D.
∴ AB = DE
Hence ΔABC ≅ ΔDEF (by SAS congruency)

Corollary: AAS (Angle-Angle-Side) congruence rule:
Two triangles are said to be congruent if two angles and one side of one triangle is equal to two angles and one side of another triangle.
In A ABC and A DEF,
∠ACB = ∠DFE . ∠ABC – ∠DEF
AB = DE
∴ ΔABC = ΔDEF (by AAS)

Given: ΔABC and ΔDEF
In which ∠A = ∠D, ∠B = ∠E
and BC = EF
To Prove: ΔABC ≅ ΔDEF
Proof: In ΔABC and ΔDEF
∠1 = ∠4 … (1) (given)
and ∠2 = ∠3 … (2) (given)
Adding eqn. (1) and (2),
∠1 + ∠2 = ∠3 + ∠4
⇒ 180° – (∠1 + ∠2) = 180° – (∠3 + ∠4) (by angle sum property)
∠ACB = ∠DFE
Hence ΔABC ≅ ΔDEF

Theorem 7.2 Angle opposite to equal sides of an isosceles triangle are equal.
If AB = AC, then
∠ABC = ∠ACB
Given: ABC is a triangle in which
AB = AC
To Prove: ∠B = ∠C


Construction:
Draw AD angle bisector of ∠A.
Proof: In ΔBAD and ΔCAD,
AB = AC (given)
∠BAD = ∠CAD (by construction)
AD = AD (common)
.∴ΔBAD = ΔCAD (by SAS)
Hence ∠B = ∠C (by CPCT)

Theorem 7.3. The sides opposite to equal angles of a triangle are equal.
Given: ΔABC in which ∠B = ∠C
To Prove: AB = AC
Construction:
Draw AD bisector of ∠A which meets BC at D.
Proof: In ΔBAD and ΔCAD,
∠B = ∠C
∠BAD – ∠CAD
AD = AD
∴ΔBAD ≅ ΔCAD
Hence AB – AC
Theorem SSS (Side-Side-Side)

Congruence rule: Two triangles are said to be congruent if all sides (three) of one triangle are equal to the all sides (three sides) of another triangles then the two triangle are congruent.
In Δ ABC and Δ DEF,
AB = DF
BC = EF
AC = DE
ΔABC ≅ΔDEF

Theorem 7.5: RHS (Right angle-Hypotenuse-Side)
Congruence rule: If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of another triangle, their the two triangles are congruent.

In ΔABC and ΔDEF,
∠ABC = ∠DEF
AC = DF
AB = DE
∴ ΔABC ≅ ΔDEF (by R.H.S)

Inequalities in a Triangle
Theorem 7.6: If two sides of a triangle are unequal, the angle opposite to the longer side is greater.
∠ABC > ∠BAC
∴ AC > BC
Given: ΔABC in which
AC > AB
To Prove:
∠ABC > ∠ACB

Construction:
Take a point D on AC such that AD = AB and join BD.
Proof: In Δ ABD,
AB – AD (by construction)
∴  ∠1 = ∠2 …(1) (Angle opposite to equal sides are equal)
∠2 > ∠3 ……..(2)  ∠1 is exterior angle of ΔBCD)

From eqn. (1) and (2),
∠1 > ∠3
Hence ∠ABC > ∠ACB.

Theorem 7.7: In a triangle, side opposite to greater angle is longer.
If ∠ABC > ∠BAC
∴ AC > BC
Given: Δ ABC in which
∠ABC > ∠ACB
To Prove: AC > AB

Proof: Here three cases arises.

• AC = AB
• AC < AB
• AC < AC

Case I: If AC = AB
∠ABC – ∠ACB (Angle opposite to equal sides are equal)
But ∠ABC > ∠ACB (given)
This is a contradiction.
Hence AC ≠AB
∴ AC > AB

Case II: AC < AB
∠ACB > ∠ABC (Angle opposite to longer side is greater)
This is contradiction of given hypothesis.
Hence only one possibility is left.
i.e. AC > AB (It must be true)
Hence AC > AB

Theorem 7.8: The sum of any two sides of a triangle is greater than the third side.
(i) AB + AC > BC
(ii) AB + BC > AC
(iii) BC + AC > AB
Given: ΔABC
Prove that:
(i) AB + BC > AC
(ii) AB + AC > BC
(iii) AC + BC > BC

Construction: BA produce to D such that AD = AC. Join CD.
Proof: In ΔACD,
AC = AD (by construction)
∠2 = ∠1 (Angle opposite to equal sides are equal)
∠2 + ∠3 > ∠1 (∠2 + ∠3 = ∠BCD)
Hence ∠BCD > ∠BDC

Hence BD > BC
⇒ AB + AC > BC (AD = AC by construction)
Similarly, AB + BC > AC
and AC + BC > AB

Median of a triangle: A line segment which joins the mid-point of the side to the opposite vertex. AD is median. D is the mid-point of BC.

Centroid: The point of intersection of all three medians of a triangle is known as its centroid.
Note: Centroid G divides the medians in the ratio 2: 1, i.e., AG : GD = 2: 1

Altitude: Perpendicular drawn from a vertex to the opposite side.

Orthocentre: The point at which all the three altitudes intersect each other is known as orthocentre.

Incentre: The point at which the bisectors of internal angles of a triangle intersect each other is called incentre.

Circumcentre: The point at which perpendicular bisectors of the sides of a triangle intersect each other is called circumcentre.

On this page, you will find Lines and Angles Class 9 Notes Maths Chapter 4 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 4 Lines and Angles will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 4 Notes Lines and Angles

Lines and Angles Class 9 Notes Understanding the Lesson

Point: A point is a dot made by a sharp pen or pencil. It is represented by capital letter.

Line: A straight and endless path on both the directions is called a line.

Line segment: A line segment is a straight path between two points.

Ray: A ray is a straight path which goes forever in one direction.

Collinear points: If three or more than three points lie on the same line, then they are called collinear points.

Non-collinear points: If three or more than three points does not lie on the same line, then they are called non-collinear points.

Angle: The space between two straight lines that diverge from a common point or between two planes that extend from a common line.

Types of Angles
1. Acute angle: An angle between 0° and 90° is called acute angle.

2. Right angle: An angle which is equal to 90° is called right angle.

3. Obtuse angle: An angle which is more than 90° but less than 180° is called obtuse angle.

4. Straight angle: An angle whose measure is 180° is called straight angle.

5. Reflex angle: An angle whose measure is between 180° and 360° is called reflex angle.

6. Complete angle: An angle which is equal to 360° is called complete angle

Pairs of Angles

1.Complementary angles: Two angles are said to be complementary if the sum of their degree measure is 90°.

For example, pair of complementary angles are 35° and 55°.

2. Supplementary angles: Two angles are said to be supplementary if the sum of their degree measure is 180°.
∠AOC + ∠BOC = 180°

3. Bisector of angle: A ray which divides an angle into two equal parts is called bisector of the angle.
∠AOC = ∠BOC

4. Adjacent angles: Two angles are said to be adjacent angles if

• They have a common vertex (O)
• They have a common arm (OC)
• and their non-common arms are on either side of common arm (OA and OB).
  ∠AOB = ∠AOC +∠BOC

5. Linear pair: Two adjacent angles are said to be linear pair if their sum is equal to 180°.

∠AOC + ∠BOC = 180°
Axiom 6.1: If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
Axiom 6.2: If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.

6. Vertically opposite angles: Vertically opposite angles are those angles which are opposite to each other (or not adjacent) when two lines cross each other.

Theorem 6.1: If two lines intersect each other, then the vertically opposite angles are equal.
To prove: If lines AB and CD mutually intersect at point O, then
(a) ∠AOC = ∠BOD (Vertically opposite angles)
(b) ∠AOD = ∠BOC

Proof: Lines AB intersect CD at O.
∠1 + ∠2 = 180° (Linear pair)
∠2 + ∠3 = 180° (Linear pair)
From eqn. (1) and (2), ∠1 + ∠2 = ∠2 + ∠3
⇒ ∠1 = ∠3 ⇒ ∠AOD = ∠BOC
Similarly, ∠AOC = ∠BOD

Parallel Lines
If distance between two lines is the same at each and every point on two lines, then two lines are said to be parallel.
If lines l and m do not intersect each other at any point then l || m.

Transversal line: A line is said to be transversal which intersect two or more lines at distinct points.

1. Corresponding angles: Pair of angles having different vertex but lying on same side of the transversal are called corresponding angles. Note that in each pair one is interior and other is exterior angle.

• ∠1 and ∠2
• ∠3 and ∠4
• ∠5 and ∠6
• ∠1 and ∠8

These angles are pair of corresponding angles.

2. Alternate interior angles: Pair of angles having distinct vertices and lying can either side of the transversal are called alternate interior angles.

• ∠1 and ∠2
• ∠3 and ∠4

These angles are alternate interior angles

3. Consecutive interior angles: Pair of interior angles of same side of transversal line.

• ∠1 and ∠2
• ∠2 and ∠4

These angles are consecutive interior angles or co-interior angles

Axiom 6.3: If two parallel lines are intersected by a transversal then each pair of corresponding angles are equal.
If AB || CD, then

• ∠PEB = ∠EFD
• ∠PEA = ∠EFC
• ∠BEF = ∠DFQ
• ∠AEF = ∠CFQ

Theorem 6.2: If two parallel lines are intersected by a transversal then pair of alternate interior angles are equal.
If AB || CD, then ?

• ∠AEF = ∠EFD
• ∠BEF = ∠CFE

Theorem 6.3: If two parallel lines are intersected by a transversal then the ! sum of consecutive interior angles of same side of transversal is equal to 180°. If AB || CD then
(i) ∠BEF + ∠DFE = 180°
(ii) ∠AEF + ∠CFE = 180°

Axiom 6.4: If two lines are intersected by a transversal and a pair of corresponding angles are equal, then two lines are parallel.
(i) If ∠PEB = ∠EFD (corresponding angles), then AB || CD

Theorem 6.4: If two lines intersected by a transversal and a pair of alternate interior angles are equal, then two lines are parallel. If ∠AEF = ∠EFD (alternate interior angles), then AB || CD.

Theorem 6.5: If two lines are intersected by a transversal and the sum of consecutive interior angles of same side of transversal is equal to 180°, the lines are parallel. If ∠AEF + ∠CFE = 180°, then AB || CD.

Theorem 6.6: Lines which are parallel to the same line are parallel to each other.
If AB || EF and CD || EF then AB || CD

Theorem 6.7: The sum of the angles of a triangle is equal to 180°.
Given: ΔABC
To prove: ∠A + ∠B + ∠C = 180°
Construction: Draw DE || BC
Proof: DE || BC
then ∠1 = ∠4 …(1) (alternate interior angles)
∠2 = ∠5 …(2) (alternate interior angles)
Adding equations (1) and (2),
∠1 + ∠2 = ∠4 +∠5
Adding ∠3 on both sides,
∠1 +∠2 + ∠3 = ∠3 + ∠4 + ∠5
⇒ ∠A + ∠B + ∠C = 180° (Sum of angles at a point on same side of a line is 180°)

Theorem 6.8: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
Given: AABC in which, side BC is produced to D.
To Prove: ∠ACD = ∠BAC + ∠ABC
Proof: ∠ACD + ∠ACB = 180° …(1) (Linear pair)
∠ABC + ∠ACB + ∠BAC = 180° …(2)
From eqn. (1) and (2), ∠ACD + ∠ACB
= ∠ABC + ∠ACB + ∠BAC
= ∠ACD = ∠ABC + ∠BAC

On this page, you will find Introduction to Euclid’s Geometry Class 9 Notes Maths Chapter 3 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 3 Introduction to Euclid’s Geometry will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 3 Notes Introduction to Euclid’s Geometry

Introduction to Euclid’s Geometry Class 9 Notes Understanding the Lesson

1. The word ‘geometry is derived from the Greek Word ‘Geo’ means Earth and ‘matrein’ means to measure,

2. In India the excavations of Harappa and Mohenjo-daro show the industrially civilisation (about 300 BCE) made use of geometry.

3. Sulbasutras were the manuals of geometrical constructions in (800 BCE to 500 BCE)

4. The Sriyantra (given in Atharvaveda) which consist 9 interwoven isosceles triangles. These triangles are arranged in such a way that they produce 43 subsidiary triangles.

5. Thales was a great mathematician who gives the proof and statement that a circle is bisected by its diameter.

6. Thales famous pupil was Pythagoaras (572 BCE). He and his group developed the theory of geometry to a great extent.

7. Euclid was a teacher of mathematics at Alexandria in Egypt collect all the famous work and arranged it in his famous treatise called ‘Elements’. He divided the elements in thirteen chapters which are each called a book. These books influenced the whole worlds to understand geometry.

Definitions which are given by Euclid

• Point-, a point is that which has no part.
• Line: A line is breadthless length.
• The ends of a line are points.
• Straight line: It is a line which lies evenly with the points on itself.
• Surface: A surface is that which has length and breadth only.
• Edge: The edges of a surface are lines.
• Plane surface: A plane surface is a surface which lies evenly with the straight lines on itself

If we study these definitions, we find the some of terms like part, length, breadth, evenly, etc. need to be further described clearly. Euclid assumed certain properties, which were not to be proved. Euclid’s assumptions are universal truths,

• Axiom: The basic facts which are taken for granted without proofs are called axiom.
• Statement: A sentence which is either true or false but both is called a statement.
• Theorem: A statement which requires proof.

Euclid’s Axioms

• Things which are equal to the same thing are equal to one another.
• If equals are added to equals, the wholes are equal.
• If equals are subtracted from equals the remainders are equal.
• Things which coincide with one another are equal to one another.
• The whole is greater than the part.
• Things which are double of the same things are equal to one another.
• Things which are halves of the same things are equal to one another.

Collinear points: Three or more points are said to be collinear, if they all lie in the same line.

Plane: A plane is a flat, two dimensional surface that extends infinitely in all directions. Intersecting lines: Two lines land m are said to be intersecting lines if l and m have only one point common.

Playfair Axiom: Two intersecting lines cannot both be parallel to a same line.

Plane figure: A figure that exist in a plane is called a plane figure.

Note:

• Common notions often called axioms.
• Postulates were the assumptions that are specific for geometry.

Axiom 5.1: There is a unique line that passes through two distinct points.

1. Through a given point infinitely many lines can be drawn.

2. A line contains infinitely many points.

Euclid’s five postulates
Postulate 1: A straight line may be drawn from any one point to any other point.
Note: This postulate tells us that one and only one (unique) line passes through two distinct points.

Postulate 2: A terminated line can be produced indefinitely.
This postulate tells us that a line segment can be extended on either side to form a line.

Postulate 3: A circle can be drawn with any centre and any radius.

Postulates 4: All right angles are equal to one another.

Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Example: Sum of ∠1 and ∠2 is less than 180°. Therefore, the lines AB and CD will enventually intersect on the left side of PQ. Nowadays, axioms and postulates are used in same sense.

Note: The statements that were proved are called propositions or theorems.
I Euclid deduced 465 propositions in a logical chain by using his axioms,

Theorem 5.1
Two distinct lines can not have more than one point in common,
Proof: Let us suppose that two lines l and m intersect in two distinct points say P and Q. Therefore two lines passing through two distinct points P and Q. But our assumption clashes with the axiom that only one line can pass through two distinct points. So, the assumption was wrong that we started with, that two lines can pass through two distinct points is wrong. Hence two distinct lines can passe through one common point.

On this page, you will find Polynomials Class 9 Notes Maths Chapter 2 Pdf free download. CBSE NCERT Class 9 Maths Notes Chapter 2 Polynomials will seemingly help them to revise the important concepts in less time.

CBSE Class 9 Maths Chapter 2 Notes Polynomials

Polynomials Class 9 Notes Understanding the Lesson

1. An expression which is the combination of constants and variables and are connected by some or all the operations addition, subtraction, multiplication and division is known as an algebraic expression.
Example: 7 + 9x – 2x² + \(\frac{5}{6}\) xy

2. Constant: Which has fixed numerical value.
Example: 7, -4, \(\frac{3}{4}\) , n etc.

3. Variable: A symbol which has no fixed numerical value is known as a variable.
Example: 2x, 5x²

4. Terms: These are the parts of an algebraic expression which are separated by operations, like addition or subtraction are known as terms.
Example: In the expression 5x³ + 9x² + 7x – 3, terms are 5x³, 9x², 7x and -3

5. Polynomial: An algebraic expression of which variables have non-negative integral powers is called a polynomial.
Example:
(a) 5x² + 7x + 3
(b) 9y³ – 7y² + 3y + 7

6. Coefficient: A coefficient is the numerical value in a term.
Note: If a term has no coefficient, the coefficient is an unwritten 1.
Example: 5x³ – 7x² – x + 3

7. Degree of a polynomial (in one variable): The highest power of the variable is called the degree of the polynomial.
Example: 5x + 4 is a polynomial in x of degree 1.

8. Degree of a polynomial in two or more variables: The highest sum of powers of variables is called the degree of the polynomial.
Example: 7x³ + 2x²y² – 3ry + 8

9. Degree of polynomial = 4 (Sum of the powers of variables x and y )

10. Types of Polynomial

(i) Linear polynomial: A polynomial of degree one is called a linear polynomial.
Example: 2x + 3 is a linear polynomial in x.

(ii) Quadratic polynomial: A polynomial of degree 2 is called a quadratic polynomial.
Example: 5x² – 7x + 4 is a quadratic polynomial.

(iii) Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial.
Example: 3x³ + 7x² – 4x + 9 is a cubic polynomial.

(iv) Biquadratic polynomial: A polynomial of degree 4 is called a biquadratic polynomial.
Example: 7x⁴ – 2x³ + 4x + 9 is a biquadratic polynomial.

11. Number of Terms in a Polynomial
Categories of the polynomial according to their terms:

(i) Moflomil A polynomial which has only one non-zero term is called a monomial.
Example: 7, 4x, \(\frac{4}{5}\) xy, 7x²y³z⁵, all are monomials.

(ii) Binomial: A polynomial which has only two non-zero terms is called binomial.
Example: 2x + 7, 9x² + 3, 3x²yz + 4x³y³z², all are binomials.

(iii) Trinomial: A polynomial which has only three non-zero terms is called a trinomial.
Example: 5x² + lx + 9, 5xy + 7xy² + 3x³yz, all are trinomials.

(iv) Constant polynomial: A polynomial which has only one term and that is a constant is called a constant polynomial.
Example: \(\frac{-3}{4}\), 7, 5 all are constant polynomials. 4
Note: The degree of constant non-zero polynomial is zero.

(v) Zero polynomial. A polynomial which has only one term i.e., 0 is called a zero polynomial.
Note: Degree of a zero polynomial is not defined.

12. Value of a Polynomial

Value of a polynomial is obtained, when variable of a given polynomial is interchanged or replaced by a ; constant.    Let p(x) is a polynomial then value of polynomial at x = a is p(a).
Zero or root of a polynomial: A zero or root of a polynomial is the value of that variable for which value of polynomial p(x) becomes zero i.e., p(x) = 0.
Let p(x) be the polynomial and x – a.
If p(a) = 0 then real value a is called zero of a polynomial.

13. Remainder Theorem
Let p(x) be a polynomial of degree ≥ 1 and a be any real number. If p(x) is divided by the linear polynomial x-a, then the remainder is p(a).

Proof: Let p(x) be any polynomial of degree greater than or equal to 1. When p(x) is divided by x – a, the quotient is q(x) and remainder is r(x).
i.e.,p(x) = (x-a) q(x) + r(x)
Since degree of x – a is 1 and the degree of r(x) is less than the degree of x – a so the degree of r(x) = 0.
It: means r(x) is a constant, say r.
Therefore, for every value of x,  r(x) = r
then   p(x) = (x-a) q(x) + r
When x = a, then  p(a) = (a – a) q(x) + r ⇒ p(a) = r

14. Factor Theorem
If p(x) is a polynomial of degree greater than or equal to 1 and a be any real number, then

• x – a is a factor of p(x) i.e., p(x) – (x-a) q(x) which shows x – a is a factor of p(x)
• Since x – a is a factor of p(x)
  p(x) = (x-a)g(x) for same polynomial g(x). In this case,p(a) = (a-a) g (a) = 0

15. Factorisation of the Polynomial ax² + bx + c by Splitting the Middle Term
Let           p(x) = ax² + bx + c
and factor of polynomial p(x) = (px + q) and (rx + s)
then   ax² + bx + c = (px + q) (rx + s) = prx² + (ps +qr)x+ qs
Comparing the coefficient of x² on both sides
a = pr …………. (1)
Comparing the coefficient of x
b =ps + qr …………. (2)
and comparing the constant terms
c = qs ……………..(3)
which shows that b is the sum of two numbers ps + qr.
Product of two numbers ps x qr =pr x qs = ac
So for factors ax² + bx + c, we should write b as sum of two numbers whose product is ac.
Example: Factorise 6x² + 17x + 5
Here,  b = p + q = 17
and   ac = 6 x 5 = 30 (= pq)
then we get factors of 30,      1 x 30, 2 x 15, 3 x 10, 5 x 6,
Among above factors of 30, the sum of 2 and 15 is 17
i.e.,p + q = 2 + 15 = 17
∴ 6x² + 17x + 5 = 6x² + (2 + 15)x + 5 = 6x² + 2x + 15x + 5
= 2x(3x + 1) + 5(3x + 1) = (3x + 1) (2x + 5)

16. Algebraic Indentities