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Factor the trinomial completely:

[tex]\[ 28x^3 + 19x^2 + 3x \][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. [tex]\( 28x^3 + 19x^2 + 3x = \square \)[/tex] (Factor completely.)

B. The polynomial is prime.

Answer :

To factor the trinomial [tex]\(28x^3 + 19x^2 + 3x\)[/tex] completely, follow these steps:

1. Identify and Factor Out the Greatest Common Factor (GCF):
Look for any common factor in all terms. Here, each term has a common factor of [tex]\(x\)[/tex]:
[tex]\[
28x^3 + 19x^2 + 3x = x(28x^2 + 19x + 3)
\][/tex]

2. Factor the Quadratic Trinomial:
Now, focus on the quadratic trinomial [tex]\(28x^2 + 19x + 3\)[/tex] inside the parentheses. We need to factor it further.

3. Use the AC Method for Factoring the Trinomial:
We need two numbers that multiply to the product of the leading coefficient (28) and the constant term (3), which is [tex]\(28 \times 3 = 84\)[/tex], and also add up to the middle coefficient, which is 19.

After checking possible pairs, we find that 12 and 7 multiply to 84 and add to 19:
[tex]\[
28x^2 + 19x + 3 = 28x^2 + 12x + 7x + 3
\][/tex]

4. Group and Factor by Grouping:
Group the terms to factor by grouping:
[tex]\[
= (28x^2 + 12x) + (7x + 3)
\][/tex]
Factor out the greatest common factor from each group:
[tex]\[
= 4x(7x + 3) + 1(7x + 3)
\][/tex]

5. Factor out the common binomial factor:
Now, restate it by factoring out the common binomial [tex]\(7x + 3\)[/tex]:
[tex]\[
= (4x + 1)(7x + 3)
\][/tex]

6. Write the Complete Factorization:
Substitute back with the factored out [tex]\(x\)[/tex] from step 1:
[tex]\[
x(28x^2 + 19x + 3) = x(4x + 1)(7x + 3)
\][/tex]

So, the complete factorization of the trinomial [tex]\(28x^3 + 19x^2 + 3x\)[/tex] is:
[tex]\[
x(4x + 1)(7x + 3)
\][/tex]

Therefore, choice A is the correct choice:
[tex]\[ 28x^3 + 19x^2 + 3x = x(4x + 1)(7x + 3) \][/tex]

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