Question

How do you factor ${n^2} - 10n + 9$?

Answer 1

Hint: In these type polynomials, we will solve by using splitting middle term, for a polynomial of the form \[a{x^2} + bx + c = 0\], rewrite the middle term as a sum of two terms whose product is \[a \cdot c = 1 \cdot 9 = 9\] and whose sum is \[b = - 10\], when you solve the expression we will get the required values.

Complete step by step solution:
The given expression is a polynomial which are algebraic expressions that are composed of two or more algebraic terms, the algebraic terms are constant, variables and exponents.
In order to factorise the expression\[a{x^2} + bx + c\], using splitting middle term, we will write the middle term as a sum of two numbers i.e., $b$ and the product of these numbers will be equal to $a \cdot c$, and we will split the middle term using these numbers, which satisfy the condition.
Now given equation is ${n^2} - 10n + 9$,
Now using the splitting the middle term condition,
Here $a = 1$ $b = - 10$ and $c = 9$, so now,
\[a \cdot c = 1 \cdot 9 = 9\] and\[b = - 10\],
Now we will factor number 9, i.e., factors of $9 = 1,3,9$,
So, we know that the sum of the numbers should be equal to $ - 10$, the numbers satisfying the condition are $ - 1$ and $ - 9$, as their sum will be equal to $ - 10$.
Now rewrite $ - 10$ as $ - 1$ and $ - 9$, then using the equation becomes,
\[ \Rightarrow {n^2} + \left( { - 1 - 9} \right)n + 9\],
Now using distributive property, we get
\[ \Rightarrow {n^2} - n - 9n + 9\],
By grouping the first two terms and last two terms, we get,
\[ \Rightarrow \left( {{n^2} - n} \right) - \left( {9n - 9} \right)\],
Now factor out the highest common factor, we get
\[ \Rightarrow n\left( {n - 1} \right) - 9\left( {n - 1} \right)\],
Now taking common term in both, we get,
\[ \Rightarrow \left( {n - 1} \right)\left( {n - 9} \right)\],
Factorising the given polynomial we get the terms \[n - 1\] and \[n - 9\].

Final Answer:
\[\therefore \]Factorising terms of ${n^2} - 10n + 9$are \[n - 1\] and \[n - 9\], and it is written as \[\left( {n - 1} \right)\left( {n - 9} \right)\].

Note:
We have several options for factoring when you are solving the polynomial equations. In a polynomial, irrespective of how many terms it has, we should always check the highest common factors first. The highest common factor is our biggest expression which would go into all our terms, and when we use H.C.F it is similar to performing the distributive property backwards. If the expression is a binomial then we must look for the differences of squares, difference of cubes, or even the sum of cubes, finally once the polynomial is factored fully, you can then use the zero property for solving the equation.