To solve the problem of subtracting the expression [tex]\((x - 8)^2\)[/tex] from 3 and expressing the result as a simplified polynomial in standard form, follow these steps:
1. Expand the expression [tex]\((x - 8)^2\)[/tex]:
The expression [tex]\((x - 8)^2\)[/tex] represents a binomial squared. To expand it, we apply the binomial expansion formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 8\)[/tex]:
[tex]\[
(x - 8)^2 = x^2 - 2 \cdot x \cdot 8 + 8^2
\][/tex]
Simplifying this, we get:
[tex]\[
(x - 8)^2 = x^2 - 16x + 64
\][/tex]
2. Subtract the expanded expression from 3:
We are asked to find the result when [tex]\((x - 8)^2\)[/tex] is subtracted from 3:
[tex]\[
3 - (x^2 - 16x + 64)
\][/tex]
To subtract the polynomial, distribute the negative sign across the terms inside the parentheses:
[tex]\[
3 - x^2 + 16x - 64
\][/tex]
3. Combine like terms:
Now, we combine the constant terms [tex]\(3\)[/tex] and [tex]\(-64\)[/tex]:
[tex]\[
3 - 64 = -61
\][/tex]
So, the resulting expression after combining the terms is:
[tex]\[
-x^2 + 16x - 61
\][/tex]
4. Express the final result in standard form:
Here, the polynomial is already in standard form, arranged in decreasing powers of [tex]\(x\)[/tex]:
[tex]\[
-x^2 + 16x - 61
\][/tex]
Thus, the simplified polynomial in standard form when [tex]\((x - 8)^2\)[/tex] is subtracted from 3 is:
[tex]\[
-x^2 + 16x - 61
\][/tex]
