Questions: Factor the trinomial, or state that the trinomial is prime. 7 x^2 - 34 x - 5

Factor the trinomial, or state that the trinomial is prime. 7 x^2 - 34 x - 5
Transcript text: Factor the trinomial, or state that the trinomial is prime. \[ 7 x^{2}-34 x-5 \]
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Solution

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Solution Steps

To factor the trinomial \(7x^2 - 34x - 5\), we need to find two numbers that multiply to \(7 \times -5 = -35\) and add up to \(-34\). If such numbers exist, we can use them to split the middle term and factor by grouping. If no such numbers exist, the trinomial is prime.

Step 1: Identify the Trinomial

We start with the trinomial: \[ 7x^2 - 34x - 5 \]

Step 2: Find Two Numbers

We need to find two numbers that multiply to \(7 \times -5 = -35\) and add up to \(-34\). These numbers are \(-35\) and \(1\).

Step 3: Split the Middle Term

We split the middle term \(-34x\) using the numbers \(-35\) and \(1\): \[ 7x^2 - 35x + x - 5 \]

Step 4: Factor by Grouping

Group the terms to factor by grouping: \[ (7x^2 - 35x) + (x - 5) \] Factor out the common factors in each group: \[ 7x(x - 5) + 1(x - 5) \]

Step 5: Factor Out the Common Binomial

Factor out the common binomial \((x - 5)\): \[ (x - 5)(7x + 1) \]

Final Answer

\[ \boxed{(x - 5)(7x + 1)} \]

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