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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
3x^2 + 22x + 7
👉

2x- 1 x + 3
? ?

3 answers

To factor the trinomial \(2x^2 + 5x - 3\), we are looking for two binomials of the form \((ax + b)(cx + d)\) that when multiplied together, result in the original trinomial.

Given the expression \(2x^2 + 5x - 3\):

  1. The product of \(a\) and \(c\) must equal the coefficient of \(x^2\), which is \(2\).
  2. The product of \(b\) and \(d\) must equal the constant term, which is \(-3\).
  3. The sum of the inner and outer products \(ad + bc\) must equal the coefficient of \(x\), which is \(5\).

After analyzing these factors, we can find that \(2x^2 + 5x - 3\) factors to:

\[ (2x - 1)(x + 3) \]

Thus, the expression \(2x^2 + 5x - 3\) is equivalent to the product of the binomials \(2x - 1\) and \(x + 3\).

For the example \(3x^2 + 22x + 7\), similar factorization methods should be applied.

Let’s represent the findings in the table format as per your example:

\[ \begin{array}{|c|c|} \hline 2x^2 + 5x - 3 & 3x^2 + 22x + 7 \ \hline (2x - 1)(x + 3) & \text{(needs to be factored)} \ \hline \end{array} \]

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

  1. The possible pairs of factors for \(3\) (product of the first term) are \(3\) and \(1\).
  2. The pairs for \(7\) (constant term) are \(7\) and \(1\).

Trial and error shows the factors of \(3x^2 + 22x + 7\) will lead to: \[ (3x + 1)(x + 7) \]

So we can complete the table with that result:

\[ \begin{array}{|c|c|} \hline 2x^2 + 5x - 3 & 3x^2 + 22x + 7 \ \hline (2x - 1)(x + 3) & (3x + 1)(x + 7) \ \hline \end{array} \]

This is your complete answer!

dumb it down

Sure! Let's break it down step-by-step in simpler terms.

For the trinomial \(2x^2 + 5x - 3\):

  1. What we're trying to do: We want to write this expression as two smaller parts (called "factors") that multiply together to give us \(2x^2 + 5x - 3\).

  2. Finding the right factors:

    • We need two numbers that:
      • When multiplied give us the last number \(-3\) (this is the constant).
      • When multiplied with the first number \(2\) to give us the middle number \(5\).

  3. The factors for \(2x^2 + 5x - 3\) can be found to be:

    • \((2x - 1)\) and \((x + 3)\).

So, we can write: \[ 2x^2 + 5x - 3 = (2x - 1)(x + 3) \]

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

  1. Work similarly: We want to find two binomials that multiply to this trinomial.

  2. Finding the right factors:

    • We need two numbers that multiply to \(3 \times 7 = 21\) (the first coefficient times the constant term) and add up to \(22\) (the middle coefficient).

  3. The factors for \(3x^2 + 22x + 7\) are found to be:

    • \((3x + 1)\) and \((x + 7)\).

So, we can write: \[ 3x^2 + 22x + 7 = (3x + 1)(x + 7) \]

Putting it all together in a table:

Here's the final result in table format:

\[ \begin{array}{|c|c|} \hline \text{Trinomial} & \text{Factors} \ \hline 2x^2 + 5x - 3 & (2x - 1)(x + 3) \ \hline 3x^2 + 22x + 7 & (3x + 1)(x + 7) \ \hline \end{array} \]

That's it! Each trinomial is broken down into its factors. If you have more questions or need clarification, feel free to ask!