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Use synthetic division to solve $\left(x^3+1\right) \div(x-1)$. What is the quotient?


A$x^2+x+1+\frac{2}{x-1}$
B$x^2+x+1+\frac{2}{x+1}$
C$x^2-x+1$
D$x^3-x^2+x$

Verified Answer :

  • Write the polynomial \(x^3 + 1\) as \(x^3 + 0x^2 + 0x + 1\).
  • Use synthetic division with the root \(c=1\) of the divisor \(x-1\).
  • Perform the division steps to obtain the quotient coefficients \(1, 1, 1\) and a remainder of \(2\).
  • Express the final result as the quotient \(x^2 + x + 1\) plus the remainder term \(\frac{2}{x-1}\), which corresponds to option (a).
  • The correct option is \(\boxed{\text{a}}\).

Explanation

  1. Setting up the Synthetic Division

To divide the polynomial \(P(x) = x^3 + 1\) by the binomial \(D(x) = x - 1\), we use synthetic division. First, express the dividend in standard form by including all missing powers of \(x\):
\[P(x) = x^3 + 0x^2 + 0x + 1.\]
The coefficients are \(1, 0, 0, 1\). Since we are dividing by \(x - 1\), we set \(c = 1\) for our synthetic division.

  1. Executing the Synthetic Division

Now, perform the synthetic division process:

  • Bring down the first coefficient, which is \(1\).
  • Multiply by \(c = 1\) to get \(1 \times 1 = 1\), and add this to the second coefficient \(0\), resulting in \(1\).
  • Multiply this sum \(1\) by \(c = 1\) to get \(1 \times 1 = 1\), and add to the third coefficient \(0\), resulting in \(1\).
  • Multiply this sum \(1\) by \(c = 1\) to get \(1 \times 1 = 1\), and add to the last coefficient \(1\), resulting in \(2\).

The sequence of values obtained is \(1, 1, 1\) as the coefficients of the quotient, and \(2\) as the remainder.

  1. Interpreting the Result and Conclusion

The quotient polynomial \(Q(x)\) is formed by the first three numbers in our sequence, which are the coefficients of \(x^2\), \(x^1\), and the constant term, giving us \(x^2 + x + 1\). The last number, \(2\), is the remainder. Therefore, the division can be expressed as:
\[\frac{x^3+1}{x-1} = x^2+x+1 + \frac{2}{x-1}.\]
Comparing this to the given options, this matches option (a).

Examples

Synthetic division is a shorthand method for polynomial division that is widely used in algebra, especially when working with roots of polynomials. For instance, when engineers analyze the stability of electronic circuits, they often need to divide transfer functions to simplify complex signal expressions. This technique allows them to quickly find factors of polynomials, which corresponds to identifying the resonant frequencies in a circuit system, ensuring it operates reliably.

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