Answer

Verified

338.4k+ views

**Hint:**

Here, we will factorize the cubic equation such that it will become a combination of a linear equation and a quadratic equation. We will find the factors of the quadratic equation by using the middle terms split method. Then by combining the factors of both the equations, we will get the factors of the given cubic equation.

**Complete step by step solution:**

We are given a cubic equation \[{x^3} - 8{x^2} + 15x\].

We will now find the factors of the given cubic equation.

First, we will take out the factor common to all the terms, so we get

\[ \Rightarrow {x^3} - 8{x^2} + 15x = x\left( {{x^2} - 8x + 15} \right)\]……………………………………………………………………..\[\left( 1 \right)\]

Now, we have a quadratic equation \[{x^2} - 8x + 15\]. So, we will find the factors of the quadratic equation.

We know that the general form of a Quadratic equation is \[a{x^2} + bx + c\].

Comparing the given quadratic equation with the general quadratic equation, we get

\[\begin{array}{l}a = 1\\b = - 8\\c = 15\end{array}\]

Now, in order to factorize the quadratic equation, we will find a pair of numbers, such that the product of the numbers is equal to the product of the first term and third term. Also, the sum of the numbers is equal to the middle terms of the equation.

Thus, the numbers \[ - 3\] and \[ - 5\] satisfies the above mentioned condition as their Product is equal to \[ac = \left( 1 \right)\left( {15} \right) = 15\] and their sum is equal to the middle terms, \[ - 8\].

Now, we will split the middle term in the given quadratic equation as \[ - 3x\] and\[ - 5x\] . So, we get

\[{x^2} - 8x + 15 = {x^2} - 3x - 5x + 15\].

By taking out the common factors, we get

\[ \Rightarrow {x^2} - 8x + 15 = x\left( {x - 3} \right) - 5\left( {x - 3} \right)\].

By grouping the common factors, we get

\[ \Rightarrow {x^2} - 8x + 12 = \left( {x - 3} \right)\left( {x - 5} \right)\].

Now, by substituting the factors of the quadratic equation in equation \[\left( 1 \right)\] , we get

\[{x^3} - 8{x^2} + 15x = x\left( {x - 3} \right)\left( {x - 5} \right)\] .

**Therefore, the factors of \[{x^3} - 8{x^2} + 15x\] are\[x\] ,\[\left( {x - 3} \right)\] and \[\left( {x - 5} \right)\].**

**Note:**

We know that the cubic equation is defined as an equation with the highest degree as 3. We know that Factorization is a process of rewriting the expression in terms of the product of the factors. Factorization is done by using the common factors, the grouping of terms and the algebraic identity. We should remember that if the sign of the product of the factors is Positive, then both the integers should be either positive or negative and the sign of the sum of the factors is Negative, then both the integers should be negative.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Which places in India experience sunrise first and class 9 social science CBSE

The list which includes subjects of national importance class 10 social science CBSE

What is pollution? How many types of pollution? Define it

State the laws of reflection of light