Distribute and Subtract Algebraic Expressions

Written as a simplified polynomial in standard form, what is the result when [tex]\((x - 8)^2\)[/tex] is subtracted from 3?

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Answer :

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]