Express (x + 9)2 as a trinomial in standard form

Express (x + 9)2 as a trinomial in standard form.

The Correct Answer and Explanation is :

To express ((x + 9)^2) as a trinomial in standard form, we will expand the expression step-by-step using the formula for squaring a binomial, which states that:

[
(a + b)^2 = a^2 + 2ab + b^2
]

In our case, (a = x) and (b = 9). We can now apply the formula:

1. Calculate (a^2):
   [
   a^2 = x^2
   ]
2. Calculate (2ab):
   [
   2ab = 2 \cdot x \cdot 9 = 18x
   ]
3. Calculate (b^2):
   [
   b^2 = 9^2 = 81
   ]

Now we can combine these results into a single expression:

[
(x + 9)^2 = x^2 + 18x + 81
]

So, the trinomial in standard form is:

[
x^2 + 18x + 81
]

Explanation

To understand the expansion process, it’s essential to recognize what squaring a binomial entails. When you square a binomial, you are multiplying the binomial by itself, which requires using the distributive property. This means each term in the first binomial multiplies each term in the second binomial.

For ((x + 9)(x + 9)), we perform the following calculations:

1. Multiply (x) by (x) to get (x^2).
2. Multiply (x) by (9) to get (9x).
3. Multiply (9) by (x) to get another (9x).
4. Multiply (9) by (9) to get (81).

Next, we combine the like terms (9x) and (9x) to get (18x). The final expression, (x^2 + 18x + 81), is in standard form, which is characterized by the order of terms from highest degree to lowest degree (in this case, (x^2), (x), and the constant).

This method of expansion is not only fundamental for algebra but also helps in various applications, such as solving quadratic equations or graphing parabolas.