Question

How do you write \[{y^2} - 3y\] in factored form?

Answer 1

Hint: Here we will factorize the given polynomial by using factor theorem. First, we will find a value of \[y\] which satisfies the polynomial by hit and trial method. Then we will take that factor and divide the polynomial by it. Finally, we will get the quotient as our second factor and desired answer.

Complete step by step solution:
We have to factorize polynomial \[{y^2} - 3y\]……\[\left( 1 \right)\]
Firstly we need to know at least one zero of the polynomial, so we will use the hit and trial method.
Let us check whether \[y = 0\] satisfies the polynomial or not
Putting \[y = 0\] in equation \[\left( 1 \right)\], we get,
\[{y^2} - 3y = 0 - 3 \times 0 = 0\]
So, \[y = 0\] is a factor as it satisfies the polynomial.
So, one of the factors is \[\left( {y - 0} \right)\].
Now we will divide the polynomial by \[y\] as,
\[\dfrac{{{y^2} - 3y}}{y} = \dfrac{{y\left( {y - 3} \right)}}{y}\]
Cancelling the similar terms, we get
\[ \Rightarrow \dfrac{{{y^2} - 3y}}{y} = \left( {y - 3} \right)\]
So we get our quotient as \[y - 3\]. Now by solving it, we get
\[\begin{array}{l}y - 3 = 0\\ \Rightarrow y = 3\end{array}\]
So, we get the zeroes of the given polynomial as \[y = 0,3\]

Hence, the factors of the polynomial will be \[\left( y \right)\left( {y - 3} \right)\]
So the factor form of \[{y^2} - 3y\] is \[\left( y \right)\left( {y - 3} \right)\].

Note:
Factor theorem states that if a polynomial \[f\left( x \right)\] of degree \[n \ge 1\] and ‘\[a\]’ is any real number then, if \[f\left( a \right) = 0\] then only \[\left( {x - a} \right)\] is a factor of the polynomial. The two problems where Factor Theorem is usually applied are when we have to factorize a polynomial and also when we have to find the roots of the polynomial. It is also used to remove known zeros from a polynomial while leaving all unknown zeros intact. It is a special case of a polynomial remainder theorem.