How to factor a trinomial when a is greater than 1

The correct Answer and Explanation is:

Factoring a trinomial of the form ( ax^2 + bx + c ), where ( a > 1 ), can be done effectively by using the method of “splitting the middle term” or “trial and error.” Here’s a step-by-step approach to factor such trinomials.

Step 1: Identify ( a ), ( b ), and ( c )

For the trinomial ( ax^2 + bx + c ), identify the coefficients ( a ), ( b ), and ( c ). For example, in the trinomial ( 6x^2 + 11x + 3 ), we have ( a = 6 ), ( b = 11 ), and ( c = 3 ).

Step 2: Multiply ( a ) and ( c )

Next, multiply ( a ) and ( c ). In our example, ( 6 \times 3 = 18 ).

Step 3: Find two numbers that multiply to ( ac ) and add to ( b )

Now, we need to find two numbers that multiply to ( ac ) (in this case, 18) and add to ( b ) (in this case, 11). The numbers that work here are 9 and 2 since ( 9 \times 2 = 18 ) and ( 9 + 2 = 11 ).

Step 4: Rewrite the trinomial

Use these two numbers to split the middle term ( bx ). Rewrite the trinomial as:
[
6x^2 + 9x + 2x + 3
]

Step 5: Factor by grouping

Now, group the terms in pairs:
[
(6x^2 + 9x) + (2x + 3)
]
Factor out the greatest common factor from each group:
[
3x(2x + 3) + 1(2x + 3)
]

Step 6: Factor out the common binomial

Now, you can factor out the common binomial ( (2x + 3) ):
[
(3x + 1)(2x + 3)
]

Conclusion

Thus, the factored form of ( 6x^2 + 11x + 3 ) is ( (3x + 1)(2x + 3) ). This method of splitting the middle term is efficient for trinomials where ( a > 1 ), as it allows for systematic identification of factors while ensuring that the final answer maintains the original polynomial’s integrity.