Question

completely factor the polynomial, if possible.

\\25x^2 + 32xy + 16y^2\\

Explanation:

Check for greatest common factor

Using the Factoring Polynomials by GCF knowledge point, we examine the terms of the trinomial:

$$ 25x^2 + 32xy + 16y^2 $$

The coefficients are \(25\), \(32\), and \(16\).
The greatest common divisor of \(25\), \(32\), and \(16\) is \(1\).
There are no common variable factors across all three terms.
Thus, the greatest common factor is \(1\).

Test for perfect square trinomial

Using the Factoring Trinomials knowledge point, we check if the expression fits the perfect square trinomial pattern:

$$ a^2 + 2ab + b^2 = (a + b)^2 $$

Identify the potential perfect square terms:

$$ 25x^2 = (5x)^2 \implies a = 5x $$

$$ 16y^2 = (4y)^2 \implies b = 4y $$

Calculate the required middle term:

$$ 2ab = 2(5x)(4y) = 40xy $$

The actual middle term is \(32xy\).
Since \(32xy
eq 40xy\), this is not a perfect square trinomial.

Attempt factoring by grouping

Using the Factoring Trinomials knowledge point, we look for two numbers that multiply to \(ac\) and add to \(b\):

$$ ac = 25 \times 16 = 400 $$

$$ b = 32 $$

We need two integers whose product is \(400\) and whose sum is \(32\).
Let's list factor pairs of \(400\) and their sums:

• \(20 \times 20 = 400 \implies 20 + 20 = 40\)
• \(10 \times 40 = 400 \implies 10 + 40 = 50\)
• \(16 \times 25 = 400 \implies 16 + 25 = 41\)
• \(8 \times 50 = 400 \implies 8 + 50 = 58\)

Since all factors are positive, the minimum sum for a product of \(400\) is \(2\sqrt{400} = 40\).
Because \(32 < 40\), no real integer factors exist that satisfy both conditions.
Thus, the polynomial cannot be factored over the integers.

Answer:

\(25x^2 + 32xy + 16y^2\) (or "No", as it is prime/not factorable)