Questions: Factor the trinomial or state that the trinomial is prime. 3x^2-13x-10

Transcript text: Factor the trinomial or state that the trinomial is prime. $3 x^{2}-13 x-10$

Solution

Solution Steps

To factor the trinomial $3x^2 - 13x - 10$, we need to find two numbers that multiply to the product of the leading coefficient (3) and the constant term (-10), which is -30, and add up to the middle coefficient (-13). Once we find these numbers, we can use them to split the middle term and factor by grouping.

Step 1: Identify the Trinomial

We start with the trinomial $3x^2 - 13x - 10$.

Step 2: Factor the Trinomial

To factor the trinomial, we find two numbers that multiply to $-30$ (the product of the leading coefficient $3$ and the constant term $-10$) and add up to $-13$. The numbers $-15$ and $2$ satisfy these conditions.

Step 3: Rewrite and Factor by Grouping

We can rewrite the trinomial as: \[ 3x^2 - 15x + 2x - 10 \] Next, we group the terms: \[ (3x^2 - 15x) + (2x - 10) \] Factoring out the common factors in each group gives us: \[ 3x(x - 5) + 2(x - 5) \] Now, we can factor out the common binomial $(x - 5)$: \[ (3x + 2)(x - 5) \]

Final Answer

The factored form of the trinomial $3x^2 - 13x - 10$ is: \[ \boxed{(3x + 2)(x - 5)} \]

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