Question

factor the trinomial, if possible.

\\x^2 + 39x + 77\\

not factorable

Explanation:

Identify the trinomial coefficients

We are given the quadratic trinomial:

$$x^2 + 39x + 77$$

This is a trinomial of the form \(ax^2 + bx + c\), where:

• \(a = 1\)
• \(b = 39\)
• \(c = 77\)

Find factors of the constant term

Using the Finding Factors with Given Sum and Product concept, we need to find two integers whose product is \(c = 77\) and whose sum is \(b = 39\).

Let's list all possible integer factor pairs of \(77\):

• \(1 \times 77 = 77\)
• \(7 \times 11 = 77\)
• \((-1) \times (-77) = 77\)
• \((-7) \times (-11) = 77\)

Check the sum of the factor pairs

Now, we calculate the sum for each factor pair to see if any pair adds up to \(39\):

• \(1 + 77 = 78

eq 39\)

• \(7 + 11 = 18

eq 39\)

• \(-1 + (-77) = -78

eq 39\)

• \(-7 + (-11) = -18

eq 39\)

Determine factorability

Using the Factoring Trinomials concept, since no pair of integer factors of \(77\) sums to \(39\), the trinomial cannot be factored over the integers. Therefore, the trinomial is not factorable.

Answer:

Not Factorable