%2F(x-4).jpeg "(x^3-13x-12)/(x-4)")
Simplifying the Expression (x^3 - 13x - 12) / (x - 4)
This expression represents a rational function, which is a fraction where both the numerator and denominator are polynomials. To simplify this expression, we can utilize polynomial long division.
Polynomial Long Division
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Set up the division:
________ x - 4 | x^3 + 0x^2 - 13x - 12 -
Divide the leading terms:
- The leading term of the divisor (x - 4) is x.
- The leading term of the dividend (x^3) is x^3.
- Divide x^3 by x, which gives x^2.
- Write x^2 above the division line.
x^2_______ x - 4 | x^3 + 0x^2 - 13x - 12 -
Multiply the quotient by the divisor:
- Multiply x^2 by (x - 4), which gives x^3 - 4x^2.
- Write the result below the dividend.
x^2_______ x - 4 | x^3 + 0x^2 - 13x - 12 x^3 - 4x^2 -
Subtract:
- Subtract the product from the dividend. Remember to change the signs of the terms you are subtracting.
x^2_______ x - 4 | x^3 + 0x^2 - 13x - 12 x^3 - 4x^2 --------- 4x^2 - 13x -
Bring down the next term:
- Bring down the next term from the dividend (-13x).
x^2_______ x - 4 | x^3 + 0x^2 - 13x - 12 x^3 - 4x^2 --------- 4x^2 - 13x - 12 -
Repeat steps 2-5:
- Divide the leading term of the new dividend (4x^2) by the leading term of the divisor (x), which gives 4x.
- Write 4x above the division line.
- Multiply 4x by (x - 4), giving 4x^2 - 16x.
- Subtract this product from the new dividend.
x^2 + 4x_____ x - 4 | x^3 + 0x^2 - 13x - 12 x^3 - 4x^2 --------- 4x^2 - 13x - 12 4x^2 - 16x --------- 3x - 12 -
Repeat steps 2-5 again:
- Divide the leading term of the new dividend (3x) by the leading term of the divisor (x), which gives 3.
- Write 3 above the division line.
- Multiply 3 by (x - 4), giving 3x - 12.
- Subtract this product from the new dividend.
x^2 + 4x + 3__ x - 4 | x^3 + 0x^2 - 13x - 12 x^3 - 4x^2 --------- 4x^2 - 13x - 12 4x^2 - 16x --------- 3x - 12 3x - 12 --------- 0 -
The result:
- The quotient is x^2 + 4x + 3.
- The remainder is 0.
Therefore, (x^3 - 13x - 12) / (x - 4) simplifies to x^2 + 4x + 3.
Conclusion
This simplification demonstrates the effectiveness of polynomial long division for dividing polynomials. The result shows that the original rational function can be expressed as a simpler polynomial, which may be useful in further analysis or calculations.