Answer

Verified

381.3k+ views

**Hint:**Here we need to use the formula where we can compare the given term by the formula:

${a^3} + {b^3} + {c^3} - 3abc = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right)$

So we can write ${x^3} - 1 + {y^3} + 3xy$ in this form and then get the required answer.

**Complete step by step solution:**

Here we are given that we need to find the factors of ${x^3} - 1 + {y^3} + 3xy$ which means we need to write it in the form of the multiplication of the two terms. So we need to see which formula is to be used.

We know that as we have the formula where we can get:

${a^3} + {b^3} + {c^3} - 3abc = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right)$

Now we can compare this formula with the given equation which is ${x^3} - 1 + {y^3} + 3xy$

Now we can write this equation as:

${x^3} + {\left( { - 1} \right)^3} + {y^3} - 3\left( { - 1} \right)xy$

If we compare this with ${a^3} + {b^3} + {c^3} - 3abc$

Then we can say that:

$

a = x \\

b = - 1 \\

c = y \\

$

Now we can simply substitute the values of all the variables of the formula with the given equation, then we will get:

${x^3} + {\left( { - 1} \right)^3} + {y^3} - 3\left( { - 1} \right)xy$$ = \left( {x - 1 + y} \right)\left( {{x^2} + {{\left( { - 1} \right)}^2} + {y^2} - x\left( { - 1} \right) - \left( { - 1} \right)y - xy} \right)$

${x^3} + {\left( { - 1} \right)^3} + {y^3} - 3\left( { - 1} \right)xy$$ = \left( {x - 1 + y} \right)\left( {{x^2} + 1 + {y^2} + x + y - xy} \right)$

Hence whenever we are given the equation and the factors are to be found then we simply need to apply the formula and then compare the terms and get the factors in the simplified form.

So we have got that:

${x^3} + {\left( { - 1} \right)^3} + {y^3} - 3\left( { - 1} \right)xy$$ = \left( {x - 1 + y} \right)\left( {{x^2} + 1 + {y^2} + x + y - xy} \right)$

**Hence we can say that A) is the correct option out of the given four options.**

**Note:**

Here the student must know the general formula of all the cubic as well as the square option. If we are given to find the factors of $\left( {{a^2} + {b^2} + 2ab - {c^2}} \right)$ then we can write it as $\left( {{{\left( {a + b} \right)}^2} - {c^2}} \right)$.

Now we can apply the formula ${x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right)$ and get the factors in simplified form.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

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

How many crores make 10 million class 7 maths CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

Write a letter to the principal requesting him to grant class 10 english CBSE