Question

factorize
${x}^{3}-6{x}^{2} + 3x + 10$
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Answer 1

༒ Given:-

A cubic Polynomial

$\bf {x}^{3} - 6 {x}^{2} + 3x + 10$

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༒ To Find

Factors of the Given Polynomial.

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༒ Concept :-

We can't factorise a cubic polynomial directly without knowing at least one factor.

We have to find that factor by hit and trial method.

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༒ Solution :-

▪Finding One factor :-)

♡Given Polynomial 》

1》 For x = 0

${0}^{3} - 6( {0)}^{2} + 3(0) + 10 \\ \\ = 10 \ne0$

2》For x = 1

${1}^{3} - 6(1) {}^{2} + 3(1) + 10 \\ \\ = 1 - 6 + 3 + 10 \\ \\ = 8 \ne0$

3》For x = -1

$( { - 1)}^{3} + - 6( { - 1)}^{2} + 3( - 1) + 10 \\ \\ = - 1 - 6 - 3 + 10 = 0$

So,

$\sf(x + 1) \: is \: a \: factor$

Now,

Dividing the given polynomial by x + 1 we get

$\sf {x}^{2} - 7x + 10$

as quotient.

Now,

Factorising the quotient:

$\sf {x}^{2} - 7x + 10 \\ \\ \sf {x}^{2} - 5x - 2x + 10 \\ \\ \sf x(x - 5) - 2(x - 5) \\ \\ \sf (x - 5)(x - 2)$

So, factors of the given cubic polynomial are

$\blue{\underline{ \boxed{\bf (x + 1)(x - 5)(x - 2)}}}$

$\Large \red{\mathfrak{  \text{W}hich \:   \: is  \:  \: the  \:  \: required }}\\ \huge \red{\mathfrak{ \text{ A}nswer.}}$

Answer 2

Given:

• x³ - 6x² + 3x + 10

To Find:

• What are the factors ?

Solution:

To solve these types of cubic equations , We have to find one factor by using trial method .

Then , we find the quotient on dividing the equation by its one found factor and then we can easily find the factors of the quotient .

Let's Begin :

Factors of 10 = 1 , 2 , 5 , 10

Let x = 1 is a factor of of polynomial

Lets check it by using Remainder theorem

→ (1)³ - 6(1)² + 3(1) + 10 = 0

→ 1 - 6 + 3 + 10 = 0

→ 11 - 6 = 0

→ 5 is not equal to 0

So, x - 1 is not a factor of polynomial

Let x = -1 is a factor of polynomial

Lets check it by using Remainder theorem

→ (-1)³ - 6(-1)² + (-3) + 10 = 0

→ - 1 - 6 - 3 + 10 = 0

→ -10 + 10 = 0

→ 0 = 0

So , x + 1 is a factor of polynomial

Now, divide the polynomial by x + 1 , We get

x² - 7x + 10 as a quotient

Now, factorize the quotient

→ x² - 7x + 10

→ x² - 5x - 2x + 10

→ x(x - 5) - 2(x - 5)

→ (x - 2) (x - 5)

We know that ,

→ x³ - 6x² + 3x + 10 = (x + 1)(x² - 7x + 10)

→ x³ - 6x² + 3x + 10 = (x + 1)(x - 2)(x - 5)

So, (x + 1)(x - 2)(x - 5) are the factors of given polynomial