Question

completely factor the trinomial, if possible.

\\x^2 - 11x + 24\\

Explanation:

Identify the trinomial coefficients

We begin with the quadratic trinomial:

$$x^2 - 11x + 24$$

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

• \(a = 1\)
• \(b = -11\)
• \(c = 24\)

Find factors with given sum and product

Using the Finding Factors with Given Sum and Product knowledge point, we need to find two integers, let's call them \(p\) and \(q\), that satisfy:

• Product: \(p \cdot q = c = 24\)
• Sum: \(p + q = b = -11\)

Since the product is positive (\(24\)) and the sum is negative (\(-11\)), both integers \(p\) and \(q\) must be negative.

Let's list the pairs of negative factors of \(24\) and find their sum:

• \(-1\) and \(-24\): sum is \(-25\)
• \(-2\) and \(-12\): sum is \(-14\)
• \(-3\) and \(-8\): sum is \(-11\)
• \(-4\) and \(-6\): sum is \(-10\)

The pair that sums to \(-11\) is \(-3\) and \(-8\).

Write the factored form

Using the Factoring Trinomials knowledge point, since the leading coefficient is \(1\), we can write the trinomial directly in its factored form using the two integers we found:

$$(x + p)(x + q)$$

Substituting \(p = -3\) and \(q = -8\):

$$(x - 3)(x - 8)$$

Verify the factorization

We can verify our result by expanding the factors using multiplication:

$$(x - 3)(x - 8) = x(x - 8) - 3(x - 8)$$

$$= x^2 - 8x - 3x + 24$$

$$= x^2 - 11x + 24$$

The expansion matches the original trinomial, confirming our factorization is correct.

Answer:

\((x - 3)(x - 8)\)