The completely factored form of P4 – 16 is (P2 + 4)(P + 2)(P – 2). This was obtained by applying the difference of squares formula twice.

To find the completely factored form of P4 – 16, we need to use a special factoring formula known as the difference of squares. This formula states that:

a2 – b2 = (a + b)(a – b)

where a and b are any real numbers.

In the case of P4 – 16, we can rewrite it as:

P4 – 16 = (P2)2 – 42

We can see that a = P2 and b = 4, so we can apply the difference of squares formula to obtain:

(P2 + 4)(P2 – 4)

However, we’re not done yet. We can still further simplify the expression by using the difference of squares formula again. This time, we have:

P2 – 4 = (P + 2)(P – 2)

Substituting this back into our original expression, we get:

P4 – 16 = (P2 + 4)(P + 2)(P – 2)

This is the completely factored form of P4 – 16.

To check that this answer is correct, we can use the distributive property of multiplication to expand the expression back out:

(P2 + 4)(P + 2)(P – 2)

= (P2 + 4) * (P2 – 2P + 2P – 4) [using the distributive property twice]

= P4 – 4P2 + 8P2 – 16

= P4 + 4P2 – 16

We can see that this expanded expression is equivalent to the original expression of P4 – 16, so we can be confident that our answer is correct.

In summary, the completely factored form of P4 – 16 is (P2 + 4)(P + 2)(P – 2).

Factoring is a mathematical process used to break down a polynomial expression into its simplest form by finding its factors. Factors are simply expressions that, when multiplied together, result in the original polynomial. For example, the factors of the polynomial x2 – 4 are (x + 2)(x – 2), because when you multiply these two expressions together, you get x2 – 4.

In the case of P4 – 16, we can see that it is a polynomial expression of degree 4, which means that it has four terms. Our goal is to find a way to break it down into a product of simpler expressions.

To do this, we use a factoring formula known as the difference of squares, which is based on the following identity:

a2 – b2 = (a + b)(a – b)

This formula tells us that the difference between the squares of two numbers can be factored into the product of their sum and difference. For example, 25 – 9 can be factored using the difference of squares formula as (5 + 3)(5 – 3) = 8 * 2 = 16.

In the case of P4 – 16, we first need to identify the squares involved. We can see that P4 is the square of P2, and 16 is the square of 4. So we can rewrite the expression as (P2)2 – 42.

We can then apply the difference of squares formula to obtain:

(P2 + 4)(P2 – 4)

This is not yet the completely factored form, as we can further simplify the expression using the difference of squares formula again. In this case, we can see that P2 – 4 is also a difference of squares, so we can factor it as:

P2 – 4 = (P + 2)(P – 2)

Substituting this back into the previous expression, we get:

(P2 + 4)(P + 2)(P – 2)

This is the completely factored form of P4 – 16, which shows us that it can be broken down into the product of three simpler expressions.

In summary, factoring is a process used to break down a polynomial expression into simpler factors. The difference of squares formula is a special factoring formula that can be used to factor expressions of the form a2 – b2. By applying this formula twice to the expression P4 – 16, we can obtain its completely factored form of (P2 + 4)(P + 2)(P – 2).