Pergunta
Perform the indicated operation: (x^3+4x^2+x-6)div (x+2) square
Solução
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MariaMestre · Tutor por 5 anos
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To perform the division of the polynomial \( \left(x^{3}+4 x^{2}+x-6\right) \) by \( (x+2) \), we will use polynomial long division.<br /><br />1. **Setup the division**:<br /> \[<br /> \frac{x^3 + 4x^2 + x - 6}{x + 2}<br /> \]<br /><br />2. **Divide the leading term of the dividend by the leading term of the divisor**:<br /> \[<br /> \frac{x^3}{x} = x^2<br /> \]<br /> Write \( x^2 \) above the division line.<br /><br />3. **Multiply \( x^2 \) by the entire divisor \( x + 2 \)**:<br /> \[<br /> x^2 \cdot (x + 2) = x^3 + 2x^2<br /> \]<br /><br />4. **Subtract this result from the original polynomial**:<br /> \[<br /> (x^3 + 4x^2 + x - 6) - (x^3 + 2x^2) = 2x^2 + x - 6<br /> \]<br /><br />5. **Repeat the process with the new polynomial \( 2x^2 + x - 6 \)**:<br /> \[<br /> \frac{2x^2}{x} = 2x<br /> \]<br /> Write \( 2x \) above the division line next to \( x^2 \).<br /><br />6. **Multiply \( 2x \) by the entire divisor \( x + 2 \)**:<br /> \[<br /> 2x \cdot (x + 2) = 2x^2 + 4x<br /> \]<br /><br />7. **Subtract this result from the current polynomial**:<br /> \[<br /> (2x^2 + x - 6) - (2x^2 + 4x) = x - 6 - 4x = -3x - 6<br /> \]<br /><br />8. **Repeat the process with the new polynomial \( -3x - 6 \)**:<br /> \[<br /> \frac{-3x}{x} = -3<br /> \]<br /> Write \( -3 \) above the division line next to \( 2x \).<br /><br />9. **Multiply \( -3 \) by the entire divisor \( x + 2 \)**:<br /> \[<br /> -3 \cdot (x + 2) = -3x - 6<br /> \]<br /><br />10. **Subtract this result from the current polynomial**:<br /> \[<br /> (-3x - 6) - (-3x - 6) = 0<br /> \]<br /><br />Since the remainder is 0, the quotient is:<br />\[<br />x^2 + 2x - 3<br />\]<br /><br />Thus, the result of the division \( \left(x^{3}+4 x^{2}+x-6\right) \div (x+2) \) is:<br />\[<br />x^2 + 2x - 3<br />\]
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