Question

How do you factor \[3{x^2} - 6x\]?

Answer 1

Hint: Factors are defined as the exact divisors of the given number; the outcome of the factors should be less than or equal to the given number. Algebraic expressions consist of numbers, which are called coefficients, and variables, which can be raised to a power. In the given expression \[3{x^2} - 6x\], 3 is the coefficient of \[3{x^2}\], the variable. Likewise, -6 is the coefficient of \[{x^1}\], hence we can find the factors by considering the common terms and evaluating the factors.

Complete step-by-step solution:
  Given,
\[3{x^2} - 6x\]
To factor an expression, you have to start by factoring out the Greatest Common Factor. List the factors of each component of the expression.
The term \[3{x^2}\]can be considered as \[3x\left( x \right)\] and the term \[6x\]can be considered as \[3x\left( 2 \right)\].
Factor \[3x\] out of \[3{x^2}\]:
\[ \Rightarrow \]\[3x\left( x \right) - 6x\]
Factor \[3x\] out of \[ - 6x\]:
\[ \Rightarrow \] \[3x\left( x \right) + 3x\left( { - 2} \right)\]
Factor \[3x\] out of \[3x\left( x \right) + 3x\left( { - 2} \right)\]:
\[ \Rightarrow \]\[3x\left( {x - 2} \right)\]
The \[3x\] is therefore a common factor that can be factored out (reversing the distributive property), and keeping the minus sign in place:
\[3{x^2} - 6x = 3x\left( {x - 2} \right)\]
This also implies that the roots (x-intercepts) of the function
\[f\left( x \right) = 3{x^2} - 6x = 3x\left( {x - 2} \right)\] are
\[x = 0\] and \[x = 2\].

Additional information: When a number is said to be a factor of any other second number, then the first number must divide the second number completely without leaving any remainder. In simple words, if a number (dividend) is exactly divisible by any number (divisor), then the divisor is a factor of that dividend.

Note: Look at the exponent’s powers of the given function 2, 1 and if you see a zero, the expression cannot be factored by a variable. When you have a binomial that is a variable with an even exponent, added to a negative number that has a square root that is a natural number, it's called a perfect square. Every number has a common factor that is one and the number itself.