Question

How do you factor ${u^3} - 1$ ?

Answer 1

To solve this question, we will apply the algebraic formula of ${a^3} - {b^3}$. We will get to factors after applying this formula. The first factor is in terms of linear equations, and the second factor is in terms of the quadratic equation. The general form of the quadratic equation is $a{x^2} + bx + c = 0$. Where ‘a’ is the coefficient of ${x^2}$, ‘b’ is the coefficient of x and ‘c’ is the constant term. Then we will try to solve the quadratic equation by sum-product pattern.
Therefore, we should follow the below steps:

Apply sum-product pattern.
Make two pairs.
Common factor from two pairs.
Rewrite in factored form.
The algebraic formula that we will use is:
${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$

Complete step-by-step answer:
In this question, we want to find the factor of ${u^3} - 1$.
Here, ${u^3} - 1$ is in the form of ${a^3} - {b^3}$ .
Here, the value of ‘a’ is u and the value of ‘b’ is 1.
As we already know the algebraic formula of ${a^3} - {b^3}$.
${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$
Now, let us apply this formula in the given expression.
For that, substitute the value of ‘a’ and ‘b’ in the above formula.
$\Rightarrow {u^3} - 1 = \left( {u - 1} \right)\left( {{u^2} + \left( 1 \right)\left( u \right) + {1^2}} \right)$
Let us simplify the right-hand side of the above equation.
$\Rightarrow {u^3} - 1 = \left( {u - 1} \right)\left( {{u^2} + u + 1} \right)$
Now, let us try to factor by splitting the middle term of ${u^2} + u + 1$.
Here, the first term is ${u^2}$ and its coefficient is 1, the middle term is u and its coefficient is 1, and the last term is 1.
Let us apply the sum-product pattern in the equation ${u^2} + u + 1$.
Since the coefficient of ${u^2}$ is 1 and the last term is 1. Let us multiply 1 and 1. The answer will be 1. We have to find the factors of 1 which sum to 1. Here, we cannot find the factor of this equation.

Hence, the answer is $\left( {u - 1} \right)\left( {{u^2} + u + 1} \right)$.

Note:
One important thing is, we can always check our work by multiplying out factors back together, and check that we have got back the original answer.
To check our factorization, multiplication goes like this:
$\Rightarrow \left( {u - 1} \right)\left( {{u^2} + u + 1} \right)$
Let us apply multiplication to remove brackets.
$\Rightarrow {u^3} + {u^2} + u - {u^2} - u - 1$
Let us simplify it. We will get,
$\Rightarrow {u^3} - 1$