To complete the polynomial equation, we need to fill in the blanks step by step. The equation we have is:
\[ (9x^2 + _x + 13) - (-2x^3 + 12x + _) = _x^2 + 5x + 2 \]
Let's rewrite this as:
\[ 9x^2 + _x + 13 + 2x^3 - 12x - _ = _x^2 + 5x + 2 \]
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Identify the leading term: The left side has a leading term of \(2x^3\), which will be the highest degree term when combined. On the right side, the leading term is a quadratic form, thus \(_x^2\) must correspond to \(2x^3\). Therefore, the leading coefficient is simply adjusted.
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Calculate coefficients:
- From the \(x\) terms, we have \(+ _ - 12 + 12 = 5\), which determines that \(_ = 5 + 12 = 17\).
- From the constant terms: \(13 - _ = 2\) gives \(_ = 11\).
Thus, as a result, we can fill in the blanks:
\[ 9x^2 + 17x + 13 - (2x^3 + 12x + 11) = 2x^2 + 5x + 2 \]
This means the completed polynomial equation is:
\[ 9x^2 + 17x + 13 \]
Minus
\(2x^3 + 12x + 11\)
Equals
\(2x^2 + 5x + 2\)
So the filled equation will be:
9x^2 + 17x + 13 - (2x^3 + 12x + 11) = 2x^2 + 5x + 2.
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