Questions: Completely factor the trinomial, if possible. x^2+42x+35

Solution Steps

We start with the trinomial:

\[ x^{2} + 42x + 35 \]

To factor the trinomial, we need to find two numbers that multiply to the product of the coefficient of \(x^{2}\) (which is 1) and the constant term (which is 35), and also add up to the coefficient of \(x\) (which is 42).

The product is:

\[ 1 \cdot 35 = 35 \]

We need two numbers that multiply to 35 and add to 42. The pairs of factors of 35 are:

• \(1 \cdot 35\)
• \(5 \cdot 7\)

None of these pairs add up to 42, indicating that the trinomial cannot be factored into linear factors with integer coefficients.

Since the trinomial \(x^{2} + 42x + 35\) cannot be factored further into simpler polynomial factors, we conclude that it is already in its simplest form.

Final Answer