%C3%B7(x-7).jpeg "(x^3-13x^2+40x+18)÷(x-7)")
Polynomial Division: (x^3 - 13x^2 + 40x + 18) ÷ (x - 7)
This article will guide you through the process of dividing the polynomial (x^3 - 13x^2 + 40x + 18) by the binomial (x - 7) using polynomial long division.
Steps of Polynomial Long Division
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Set up the division problem: Write the dividend (x^3 - 13x^2 + 40x + 18) inside the division symbol and the divisor (x - 7) outside.
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Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2, which we write above the division symbol.
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Multiply the divisor by the quotient term: Multiply (x - 7) by x^2, resulting in x^3 - 7x^2. Write this result below the dividend.
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Subtract: Subtract the result from step 3 from the dividend. This leaves us with -6x^2 + 40x + 18.
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Bring down the next term: Bring down the next term from the dividend (40x).
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Repeat steps 2-5: Now, divide the new leading term (-6x^2) by the leading term of the divisor (x), which gives us -6x. Write this term above the division symbol. Multiply (x - 7) by -6x, resulting in -6x^2 + 42x. Subtract this from the current expression to get -2x + 18.
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Bring down the next term: Bring down the last term from the dividend (18).
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Repeat steps 2-5: Divide the new leading term (-2x) by the leading term of the divisor (x), which gives us -2. Multiply (x - 7) by -2, resulting in -2x + 14. Subtract this from the current expression, leaving us with 4.
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Remainder: The final result is 4, which is the remainder.
Conclusion
Therefore, the result of dividing (x^3 - 13x^2 + 40x + 18) by (x - 7) is:
(x^3 - 13x^2 + 40x + 18) ÷ (x - 7) = x^2 - 6x - 2 + 4/(x - 7)
This means the quotient is x^2 - 6x - 2 and the remainder is 4.