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What are the factors of x3 – 13x2 – 30x?

  • Last Updated : 24 Sep, 2021

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 simplify 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

Basic rules of Exponents

There are certain basic rules defined for exponents in order to solve the exponential expressions along with the other mathematical operations, for example, if there are the product of two exponents, it can be simplified to make the calculation easier and is known as product rule, let’s look at some of the basic rules of exponents,

  • Product Rule ⇢ an + am = an + m
  • Quotient Rule ⇢ an / am = an – m
  • Power Rule ⇢ (an)m = an × m or m√an = an/m
  • Negative Exponent Rule ⇢ a-m = 1/am
  • Zero Rule ⇢ a0 = 1
  • One Rule ⇢ a1 = a

What is a Factor?

A factor of a number divides that number completely without leaving any remainder. For instance, take a number 30, now 30 have plenty of factors and they are the numbers that leave no remainder. Factors are 2, 3, 5, 6, 10, 15, 30. Another example can be the number 10, factors of 10 are 2, 5, 10. 1 and the number itself are always the factors of the number. Let’s look at the problem statement now,

What are the factors of x3 – 13x2 – 30x?

Solution:

Take x common from the expression,

= x (x2 – 13x – 30)

Now, factorize the expression inside the bracket,

= x (x2 – 15x + 2x – 30)

= x (x (x – 15) + 2 (x – 15))

= x (x + 2) (x – 15)

Equate this expression to 0 in order to find out factors,

x (x + 2) (x – 15) = 0

Therefore, x = 0, x = -2, x = 15

Similar Problems

Question 1: Factorize x2 – 400.

Solution:

Using exponents, write x2 – 400 as x2 – 202

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

= (x2 – 202)

= (x + 20)(x – 20)

Therefore, x = -20, x = 20

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 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

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