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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

3 answers

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:

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

wrong

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!