Question

Factors of $ {q^2} - 10q + 21 $ are:
A. $ \left( {q - 3} \right) $
B. $ \left( {q - 3} \right) $
C. $ \left( {q - 7} \right) $
D.0

Answer 1

Hint: The given expression can be factorised by using the identity of $ {\left( {a - b} \right)^2} $ if the end terms of the expression are the perfect square otherwise they can be factorised by splitting the middle term of the expression.

Complete step-by-step answer:
The given expression for the factorization is: $ {q^2} - 10q + 21 $
Here we see the end terms of the given expression which are $ {q^2} $ and 21. So we found that $ {q^2} $ is the perfect square of q but 21 is not the perfect square.
Therefore, we have to use the splitting method to solve the expression.
We have to replace the middle term 10 or we have to distribute the middle term 10 in two parts so that their addition will be equal to 10 and their multiplication is equal to the 21.
So 3 and 7 are the number or digit which can be the middle term 10 then we get,
 $
 \Rightarrow {q^2} - \left( {7 + 3} \right)q + 21\\
 \Rightarrow {q^2} - 7q - 3q + 21
$
Here we see that q is common between the first two terms of the expression and 3 is common between the last two terms of the expression. So carry out these common then we get,
$ \Rightarrow q\left( {q - 7} \right) - 3\left( {q - 7} \right) $
Take common $ \left( {q - 7} \right) $ from the expression and the remaining expression in the other bracket.
$ \Rightarrow \left( {q - 3} \right)\left( {q - 7} \right) $
So, this is the required factor of the given expression.
Hence, option A $ \left( {q - 3} \right) $ is correct as well option C $ \left( {q - 7} \right) $ is also correct.

So, the correct answer is “Option A AND C”.

Note: The factor of the quadratic expression cube carried out by using the identities and also by using the splitting middle term method. Identities can be applicable for equations of higher degree but the splitting method is only applicable when the expression has quadratic form.