Question

Which expression is equivalent to the trinomial?
Type your answers to each problem in the table below. The order does not matter!



Example
2x^2 + 5x - 3 👉 2x- 1 x + 3
3x^2 + 22x + 7

Answer 1

wrong

Answer 2

To factor the trinomial \(3x^2 + 22x + 7\), we need to find two numbers that multiply to the product of the coefficient of \(x^2\) (which is 3) and the constant term (which is 7). This product is \(3 \cdot 7 = 21\). We also need these two numbers to add up to the coefficient of \(x\), which is 22.

The two numbers that satisfy these conditions are 21 and 1, since \(21 \times 1 = 21\) and \(21 + 1 = 22\).

Now, we can rewrite the middle term (22x) using these two numbers:

\[
3x^2 + 21x + 1x + 7
\]

Next, we group the terms:

\[
(3x^2 + 21x) + (1x + 7)
\]

Now, we can factor by grouping:

1. From the first group \(3x^2 + 21x\), we can factor out \(3x\):
\[
3x(x + 7)
\]

2. From the second group \(1x + 7\), we can factor out 1:
\[
1(x + 7)
\]

Now we have:
\[
3x(x + 7) + 1(x + 7)
\]

We can now factor out \((x + 7)\):
\[
(3x + 1)(x + 7)
\]

Thus, the expression equivalent to \(3x^2 + 22x + 7\) is:

\[
(3x + 1)(x + 7)
\]

So the final answer is:

\[
3x + 1 \quad \text{and} \quad x + 7
\]

Answer 3

Apologies for any confusion earlier. Let's try factoring \(3x^2 + 22x + 7\) step by step again.

To factor the trinomial \(3x^2 + 22x + 7\):

1. **Identify** A, B, and C: In the expression \(Ax^2 + Bx + C\), we have:
- \(A = 3\)
- \(B = 22\)
- \(C = 7\)

2. **Multiply A and C**: We need two numbers that multiply to \(3 \times 7 = 21\) and add up to \(22\).

3. **Find the two numbers**: The two numbers that meet these criteria are \(21\) and \(1\). These numbers multiply to \(21\) and add to \(22\).

4. **Rewrite the expression**: Now we can rewrite \(22x\) using \(21x\) and \(1x\):
\[
3x^2 + 21x + 1x + 7
\]

5. **Group the terms**:
\[
(3x^2 + 21x) + (1x + 7)
\]

6. **Factor out the common factors**:
- From the first group, factor out \(3x\):
\[
3x(x + 7)
\]
- From the second group, factor out \(1\) (or just leave it):
\[
1(x + 7)
\]

7. **Combine the factors**:
\[
3x(x + 7) + 1(x + 7) = (3x + 1)(x + 7)
\]

Thus, the correct factors of the trinomial \(3x^2 + 22x + 7\) are:
\[
(3x + 1)(x + 7)
\]

These two expressions are indeed the correct answer, and I apologize for any earlier confusion. If you meant to look for a specific form or format, please clarify. Thank you for your patience!