Questions: Completely factor the trinomial, if possible. 3x^2-25x+28

Completely factor the trinomial, if possible. 3x^2-25x+28
Transcript text: Completely factor the trinomial, if possible. \[ 3 x^{2}-25 x+28 \]
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Solution

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Solution Steps

To factor the trinomial \(3x^2 - 25x + 28\), we need to find two numbers that multiply to the product of the leading coefficient (3) and the constant term (28), which is 84, and add up to the middle coefficient (-25). Once these numbers are found, we can use them to split the middle term and factor by grouping.

Step 1: Identify the Trinomial

We start with the trinomial \(3x^2 - 25x + 28\).

Step 2: Factor the Trinomial

To factor the trinomial, we find two numbers that multiply to \(3 \times 28 = 84\) and add up to \(-25\). The numbers \(-21\) and \(-4\) satisfy these conditions.

Step 3: Rewrite and Factor by Grouping

We can rewrite the trinomial as: \[ 3x^2 - 21x - 4x + 28 \] Next, we group the terms: \[ (3x^2 - 21x) + (-4x + 28) \] Factoring each group gives us: \[ 3x(x - 7) - 4(x - 7) \] Now, we can factor out the common term \((x - 7)\): \[ (3x - 4)(x - 7) \]

Final Answer

The completely factored form of the trinomial \(3x^2 - 25x + 28\) is: \[ \boxed{(3x - 4)(x - 7)} \]

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