To divide the polynomials \( 8x^3 + 22x^2 - 63x + 9 \) by \( x^2 \), we'll use long division.
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Set up the division: \[ \begin{array}{r|rr} x^2 & 8x^3 + 22x^2 - 63x + 9 \ \end{array} \]
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Divide the leading term: Divide \( 8x^3 \) by \( x^2 \) to get \( 8x \).
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Multiply and subtract: Multiply \( 8x \) by \( x^2 \) to get \( 8x^3 \), then subtract: \[ (8x^3 + 22x^2 - 63x + 9) - (8x^3) = 22x^2 - 63x + 9 \]
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Repeat the process: Now, divide the leading term \( 22x^2 \) by \( x^2 \) to get \( 22 \).
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Multiply and subtract again: Multiply \( 22 \) by \( x^2 \) to get \( 22x^2 \), then subtract: \[ (22x^2 - 63x + 9) - (22x^2) = -63x + 9 \]
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Continue the process: Now we divide the leading term \(-63x\) by \(x^2\) but since \(-63x\) is of lower degree than \(x^2\), the division ends here.
Putting it all together, the result of the polynomial long division is: \[ 8x + 22 + \frac{-63x + 9}{x^2} \]
So, the final result can be expressed as: \[ 8x + 22 - \frac{63x - 9}{x^2} \]
This represents the quotient \( 8x + 22 \) with a remainder of \( -63x + 9 \).
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