Question

How do you factor ${{y}^{3}}-{{y}^{2}}+y-1$ by grouping?

Answer 1

Hint: First let us take-out common terms from the first two terms and last two terms. Then write them together as the product of sums form. Now among the factorized terms, find the expression which can be further factorized and then write that also in the product of sums form. Now write all of them together and represent them as the factors of the given expression.

Complete step-by-step solution:
The given polynomial which must be factored is ${{y}^{3}}-{{y}^{2}}+y-1$
The polynomial is of degree $3$
To factorize this polynomial, we shall,
Now, firstly let us take the common terms out of the first two terms.
We consider two terms together because there are four terms in total and to factorize, we need to group them into two halves.
$\Rightarrow {{y}^{2}}\left( y-1 \right)+y-1$
Now secondly take the common terms out of the last two terms.
Since there is nothing common to take out, we shall take one as the common factor.
$\Rightarrow {{y}^{2}}\left( y-1 \right)+1\left( y-1 \right)$
Now on writing it in the form of the product of sums, also known as factoring,
$\Rightarrow \left( {{y}^{2}}+1 \right)\left( y-1 \right)$
Now writing it all together we get the factors as,
$\Rightarrow \left( {{y}^{2}}+1 \right)\left( y-1 \right)$
Hence the factors for the polynomial ${{y}^{3}}-{{y}^{2}}+y-1$ are $\left( {{y}^{2}}+1 \right)\left( y-1 \right)$.

Note: The process of factorization is reverse multiplication. In the above question, we have multiplied three linear line equations to get a $3$ degree equation (a polynomial of degree $3$ ) expression using the distributive law. Whenever there is a polynomial that is to be solved, the solution contains the roots of the expression. The number of roots is decided by the degree of the polynomial.
Whenever there is an odd number of terms in the expression, always factorize the middle term. If there are even numbers of terms in the expression, group them together in halves and then find the common factor amongst them and take it out. This process is known as grouping.