Write $3(x - 2)^2 - 3(x - 2) + 8$ as a simplified polynomial in standard form.
Understand the Problem
The question asks us to simplify the given polynomial expression and write it in standard form. This involves expanding the squared term, distributing the constants, combining like terms, and arranging the terms in descending order of their exponents.
Answer
$3x^2 - 15x + 26$
Answer for screen readers
$3x^2 - 15x + 26$
Steps to Solve
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Expand the squared term Expand $(x-2)^2$ using the formula $(a-b)^2 = a^2 - 2ab + b^2$: $(x-2)^2 = x^2 - 2(x)(2) + 2^2 = x^2 - 4x + 4$
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Distribute the first constant Multiply the result from step 1 by 3: $3(x^2 - 4x + 4) = 3x^2 - 12x + 12$
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Distribute the second constant Multiply $(x - 2)$ by $-3$: $-3(x - 2) = -3x + 6$
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Combine the results Add the results from steps 2 and 3, along with the constant 8: $(3x^2 - 12x + 12) + (-3x + 6) + 8$
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Combine like terms Combine the $x^2$, $x$, and constant terms: $3x^2 + (-12x - 3x) + (12 + 6 + 8) = 3x^2 - 15x + 26$
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Write in standard form The polynomial is already in standard form, which means the terms are arranged in descending order of their exponents: $3x^2 - 15x + 26$
$3x^2 - 15x + 26$
More Information
The polynomial $3x^2 - 15x + 26$ is a quadratic expression, which is a polynomial of degree 2. Standard form means the terms are written from highest to lowest degree.
Tips
A common mistake is not distributing the negative sign correctly when expanding $-3(x-2)$, which would lead to an incorrect $x$ term, and also error in calculating the constant. Another common mistake is forgetting to expand $(x-2)^2$, which can lead to an incorrect quadratic term.
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