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Factorize x3 + 4x2 – 9x – 36.

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Mathematics is not only about numbers but it is about dealing with different calculations involving numbers and variables. This is what basically is known as Algebra. Algebra is defined as the representation of calculations involving mathematical expressions that consist of numbers, operators, and variables. Numbers can be from 0 to 9, operators are the mathematical operators like +, -, ×, ÷, exponents, etc, variables like x, y, z, etc.

Exponents and Powers

Exponents and powers are the basic operators used in mathematical calculations, exponents are used to simplifying the complex calculations involving multiple self multiplications, self multiplications are basically numbers multiplied by themselves. For example, 7 × 7 × 7 × 7 × 7, can be simply written as 75. Here, 7 is the base value and 5 is the exponent and the value is 16807. 11 × 11 × 11, can be written as 113, here, 11 is the base value and 3 is the exponent or power of 11. The value of 113 is 1331.

Exponent is defined as the power given to a number, the number of times it is multiplied by itself. If an expression is written as cxy where c is a constant, c will be the coefficient, x is the base and y is the exponent. If a number say p, is multiplied n times, n will be the exponent of p. It will be written as

p × p × p × p … n times = pn

Factorize x3  + 4x2 – 9x – 36.

Solution:

The expression has a cubic term and therefore, cannot be factorized directly. Let’s break the expression into two parts and observe if some terms can be taken out. Here, the expression x3 + 4x2 – 9x – 36 can be broken as,

  • x3 + 4x2
  • – 9x – 36

Taking x2 common from the first part and -9 common from the second part. The expression shall look like this,

= x2 (x + 4) – 9 (x + 4)

= (x2 – 9)(x + 4)

= (x2 – 32)(x + 4)

Using identity, (x2 – y2) = (x + y)(x – y)

= (x + 3)(x – 3)(x + 4)

Therefore, x = -3, x = 3, x = -4

Similar Problems

Question 1: Factorize, x2 – 16.

Solution:

Using exponents, write x2 – 16 as x2 – 42

Using identity, (x2 – y2) = (x + y)(x – y)

= (x2 – 42

= (x + 4)(x – 4)

Therefore, x = -4, x = 4

Question 2: Factorize, x3 + 7x2 – 7x – 49.

Solution:

The expression x3 + 7x2 – 7x – 49 can be broken as,

  • x3 + 7x2
  • -7x – 49

Taking x2 common from the first part and -7 common from the second part. The expression shall look like this,

= x2 (x + 7) – 7 (x + 7)

= (x2 – 7)(x + 7)

= (x2 – (√7)2)(x + 7)

Using identity, (x2 – y2) = (x + y)(x – y)

= (x + √7)(x – √7)(x + 7)

Therefore, x = -√7, x = √7, x = 7

Question 3: Factorize, x3 – 9x2 + 6x – 54.

Solution:

The expression x3 – 9x2 + 6x – 54 can be broken as,

  • x3 – 9x2
  • 6x – 54

Taking x2 common from the first part and +6 common from the second part. The expression shall look like this,

= x2 (x – 9) – 7 (x – 9)

= (x2 – 9)(x + 9)

= (x2 – (3)2)(x + 9)

Using identity, (x2 – y2) = (x + y)(x – y)

= (x + 3)(x – 3)(x + 7)

Therefore, x = -3, x = 3, x = -7


Last Updated : 25 Dec, 2023
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