Class 9 Maths

# Polynomials

## Exercise 2.4 Part 4

Question 4: Factorise:

(i) 12x^2 - 7x + 1

Answer: By Splitting the middle term method:
Step – 1 – Multiply coefficient of x2 i.e. 12 and constant term 1
By multiplication of 12 and 1 we get 12
Step – 2 – Find out the probable prime factors of 12
Prime factors of 12 are (2 and 6) and (3 and 4)

Step 3 – Now take that prime factor, sum of which becomes equal to middle term of the given polynomial, i.e. coefficient of x which is equal to – 7
Here, 2 + 6 ≠ 7
3 + 4 = 7
So, we will consider the factor (3 and 4) of 12
Now, arrange the given polynomial as given below:
Given, 12x^2 - 7x + 1
= 12x^2 - (3x + 4x) + 1
(Because 3x + 4x = 7x)
= 12x^2 - 3x – 4x + 1
By taking 3x and -1 as common, we get:
= 3x(4x – 1) – 1(4x – 1)
By taking (4x – 1) as common, we get:
= (3x – 1)(4x – 1)

Factors of given polynomial = (3x – 1) and (4x – 1)

(ii) 2x^2 + 7x + 3

Answer: Given, 2x^2 + 7x + 3
Coefficient of x^2 = 2
Constant term = 2
Thus, ab = 6
Prime factors of 6 = 1 and 6
Now, given polynomial can be written as follows:
2x^2 + x + 6x + 3
= x(2x + 1) + 2(2x + 1)
= (2x + 1)(x + 3)

Thus, factor of given polynomial = (2x + 1)(x + 3)

(iii) 6x^2 + 5x – 6

Answer: Given, 6x^2 + 5x – 6

Here, a = 6 and b = 6
Therefore, ab = 36
Factor of 36 = 9 and 4
Thus, given polynomial can be arranged as follows:
6x^2 + 9x – 4x – 6
= 3x(2x + 3) – 2(2x + 3)
= (2x + 3)(3x – 2)

(iv) 3x^2 - x – 4

Answer: Given, 3x^2 - x – 4
Here, ab = 12
Factor of 12 = 3 and 4
Thus, given polynomial can be arranged as follows:
3x^2 + 3x – 4x – 4
= 3x(x + 1) – 4(x + 1)
= (x + 1)(3x – 4)

Thus, factor of given polynomial = (x + 1)(3x - 4)

## Number System

Numbers are in the form of p/q or can be written in the form of p/q are called rational numbers.

## Polynomials

Algebraic expression with many terms having variables and coefficients is called polynomial.

## Coordinate Geometry

The distance of a point from the y - axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.

## Linear Equations in Two Variables

A linear equation in two variables has infinite number of solutions.

## Euclid's Geometry

The basic concept about a line is that it should be straight and that it should extend indefinitely in both the directions.

## Lines & Angles

An angle is formed by two rays originating from the same end point. The rays making an angle are called the arms of the angle and the end-points are called the vertex of the angle.

## Triangles

Two figures are congruent, if they are of the same shape and of the same size. Two circles of the same radii are congruent. Two squares of the same sides are congruent.

Sum of the angles of a quadrilateral is 360°. A diagonal of a parallelogram divides it into two congruent triangles.

## Parallelograms

In a parallelogram: opposite sides are equal, opposite angles are equal, and diagonals bisect each other

## Circles

The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle.

## Heron's Formula

Heron's formula is generally used for calculating area of scalene triangle.

## Surface Area

How to calculate surface area of cylinder, cone and sphere?

## Volume

How to calculate volume of cylinder, cone and sphere?

## Statistics

Learn to make bar chart and histogram from given data.

## Probability

Probability tells about likelihood of an event to occur.