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How do you factor 7x2+5x2?

Answer
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Hint: For factoring polynomials, “factoring” (or “factoring completely”) is always done using some set of numbers as a possible coefficient. When all the factors are linear, it is clear that the expression is factored completely. If we have an irrational factor then we will definitely have its conjugate form also.

Complete step by step answer:
As per the given question, we have to factor 7x2+5x2 completely.
Firstly, we multiply the coefficient of x2 with the constant term -2. That is, we get 7×2=14.
As the product is negative, then we have to choose two numbers whose sum (or difference) is coefficient of x and product of these two numbers must be equal to -14.
Finding the numbers: As we know that the sum of 7 and -2 gives the coefficient of x (that is, 5) and their product is -14.
72=57×2=14
So, we can say that the numbers are 7 and -2.
Now, we substitute these values into the expression. Then, we get
7x2+7x2x2 ( Notice that we replaced the 5 with sum of 7 and -2)
Factoring the first two expressions, we can take 7x as common. Then we can write as
7x2+7x7x(x+1)
Factoring the next two expressions, we can take -2 as common. Then we can write as
2x22(x+1)
From above, we notice that (x+1) is the same in both. Therefore, we can rewrite the given expression as:
7x(x+1)2(x+1)(7x2)(x+1)
Here, we can see that factors are linear. Thus, this is the completely factored form of the given expression 7x2+5x2.
We can always check by expanding:
(7x2)(x+1)=7x2+7x2x27x2+5x2 which is the original. So, we are correct.

On factoring the expression 7x2+5x2 completely, we get (7x2)(x+1).

Note: While solving these types of problems, we generally make a mistake leaving the complex factors like (x2+4) unfactorized. We have to factor these ones also. And, we need to have enough knowledge on the methods of factoring an expression completely. It is better if we avoid calculation or simplification mistakes.
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