what are the factor of the following trinomial?
3t^2 + 19t + 6
a) (t + 1)(3t + 6)
b) (3t + 1)(t + 6)
c) (t - 1) (3t + 6)
d) (3t - 1) (t + 6)
If we note that 19=3*6+1*1, we would choose the answer where all signs are positive, and which gives the term 18t².
Look at answer (b).
Step 1.
Multiply 3t by t
Step 2.
Multiply 3t by 6
Step 3.
Multiply 1 by t
Step 4.
Multiply 1 by 6
That gives you
3t^2+18t+t+6
Step 5.
End Result
3t^2 + 19t + 6
To find the factors of the trinomial 3t^2 + 19t + 6, we can use the factoring method. Here's how you can do it:
Step 1: Multiply the coefficient of the quadratic term (3) by the constant term (6). In this case, 3 * 6 = 18.
Step 2: Find two numbers that multiply to give the result from Step 1 (18) and add up to the coefficient of the linear term (19). In this case, the numbers are 3 and 6 because 3 * 6 = 18 and 3 + 6 = 9, which is the coefficient of the linear term (19).
Step 3: Rewrite the trinomial using the numbers from Step 2. The quadratic term remains the same. Replace the linear term (19t) with the two numbers we found (3t and 6t). So, the trinomial becomes:
3t^2 + 3t + 6t + 6
Step 4: Group the terms in pairs, and factor by grouping:
(3t^2 + 3t) + (6t + 6)
3t(t + 1) + 6(t + 1)
Step 5: Factor out the common binomial factor (t + 1):
(t + 1)(3t + 6)
Therefore, the factorization of the trinomial 3t^2 + 19t + 6 is (t + 1)(3t + 6).
Comparing this factorization with the given options:
a) (t + 1)(3t + 6) - This matches the factorization we found.
b) (3t + 1)(t + 6) - This does not match the factorization we found.
c) (t - 1) (3t + 6) - This does not match the factorization we found.
d) (3t - 1) (t + 6) - This does not match the factorization we found.
Therefore, the correct answer is option a) (t + 1)(3t + 6).