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Question: Factorise: \( 84 - 2r - 2{r^2} \) (A) \( 2\left( {7 + r} \right)\left( {6 - r} \right) \) (B) ...

Factorise: 84−2r−2r284 - 2r - 2{r^2}
(A) 2(7+r)(6−r)2\left( {7 + r} \right)\left( {6 - r} \right)
(B) (7−r)(6−r)\left( {7 - r} \right)\left( {6 - r} \right)
(C) −2(7+r)(6−r)- 2\left( {7 + r} \right)\left( {6 - r} \right)
(D) (7−r)(6+r)\left( {7 - r} \right)\left( {6 + r} \right)

Explanation

Solution

Hint : We have been given a quadratic algebraic expression to factorize. Factoring an expression means finding the factors of the given expression and writing the expression in the form of the product of the factors. A factor is a number or an expression which divides the given expression completely. A given expression can have any number of factors.

Complete Step By Step Answer:
We have been given to factorize the expression 84−2r−2r284 - 2r - 2{r^2} . Factoring the expression means finding the factors.
We can rearrange the terms and write it as,
−2r2−2r+84- 2{r^2} - 2r + 84
Since the highest power of the variable is 22 , the given expression is a quadratic expression.
First we see that we can take 22 common from all the terms. So 22 is one of the factors.
We can write,
−2r2−2r+84=2(−r2−r+42)- 2{r^2} - 2r + 84 = 2\left( { - {r^2} - r + 42} \right)
We will use the following method to find the factors of the quadratic expression wherein we will split the middle term.
If the given expression is ar2+br+ca{r^2} + br + c , then we split bb into pp and qq such that,
p+q=bp + q = b
and pq=acpq = ac
We can write −1=−7+6- 1 = - 7 + 6
Thus,
2(−r2−r+42)=2(−r2−7r+6r+42)2\left( { - {r^2} - r + 42} \right) = 2\left( { - {r^2} - 7r + 6r + 42} \right)
Now we can take −r- r common from the first two terms and 66 common from the last two terms.
We have,
2(−r2−7r+6r+42)=2(−r(r+7)+6(r+7))2\left( { - {r^2} - 7r + 6r + 42} \right) = 2\left( { - r\left( {r + 7} \right) + 6\left( {r + 7} \right)} \right)
We can further simplify as,
2(−r(r+7)+6(r+7))=2(r+7)(−r+6)=2(7+r)(6−r)2\left( { - r\left( {r + 7} \right) + 6\left( {r + 7} \right)} \right) = 2\left( {r + 7} \right)\left( { - r + 6} \right) = 2\left( {7 + r} \right)\left( {6 - r} \right)
Thus, we get the factored form of the given expression as 2(7+r)(6−r)2\left( {7 + r} \right)\left( {6 - r} \right) .
Hence, option (A) is the correct answer.

Note :
Factors of an expression can be a number or an expression. When we are finding factors for an expression, the factors should be such that it cannot be factored further. We have to be careful about the conditions while splitting the middle term. We can check that when we multiply the three resulting factors we will end up with the given expression.