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# How do you factorize${x^2} - 3x$.

Last updated date: 22nd Jun 2024
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Hint: Here we will factorize the given polynomial by using the factor theorem. First, we will find a value of $x$ which satisfies the polynomial by hit and trial method. Then we will take that factor and divide the polynomial by it, to find the next set of factors.

We have to factorize polynomials ${x^2} - 3x$.
First, we need to know at least one zero of the polynomial, so we will use the hit and trial method.
Let us check whether $x = 0$ satisfies the polynomial or not.
Putting $x = 0$ in ${x^2} - 3x$, we get,
${0^2} - 3 \times 0 = 0$
So $x = 0$ is a factor as it satisfies the polynomial.
So, one factor is $\left( {x - 0} \right)$ or we can say $x$ is the factor of the given polynomial.
Now we will divide the polynomial ${x^2} - 3x$ by $x$ and take out the common term from both the terms of the polynomial. Therefore, we get
$\dfrac{{{x^2} - 3x}}{x} = \dfrac{{x\left( {x - 3} \right)}}{x}$
Cancelling out the common terms, we get
$\Rightarrow \dfrac{{{x^2} - 3x}}{x} = \left( {x - 3} \right)$
So, we get the quotient as $\left( {x - 3} \right)$.
So, we get the zeroes of the given polynomial as $x = 3$.

Hence, the factors of the polynomial are $\left( x \right)\left( {x - 3} \right)$

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.