How to use synthetic division to find zeros?
Understand the Problem
The question is asking about the method of synthetic division and how it can be used to find the zeros of a polynomial function. Synthetic division is a simplified form of polynomial long division which can help in dividing a polynomial by a linear factor, thus determining its roots.
Answer
Use synthetic division to divide the polynomial by $x - r$; if the remainder is zero, $r$ is a zero.
Answer for screen readers
The zeros of the polynomial can be found using synthetic division, and if the division yields a remainder of zero, that root $r$ is a zero of the polynomial.
Steps to Solve
- Identify the polynomial and the linear factor
Start by determining the polynomial function you want to divide and the linear factor (in the form $x - c$) for which you want to find the zeros. Let's say our polynomial is $P(x) = ax^3 + bx^2 + cx + d$ and we want to divide by $x - r$.
- Set up synthetic division
Write down the coefficients of the polynomial. For $P(x) = ax^3 + bx^2 + cx + d$, you will need the coefficients $[a, b, c, d]$. Then, based on the zero you are checking, $r$, write down the value of $r$ on the left side.
- Perform synthetic division steps
Bring down the first coefficient (the leading coefficient).
Next, multiply $r$ by the number you just brought down and add this result to the next coefficient. Repeat this process for all coefficients.
- Interpret the result
The final row gives you the coefficients of the quotient polynomial. The last number you obtain is the remainder. If the remainder is zero, then $x - r$ is a factor of the polynomial, indicating that $r$ is a zero of the polynomial.
- Verify if $r$ is a zero
If the remainder is zero, confirm that you have correctly found a root by substituting $r$ back into the original polynomial $P(x)$. If $P(r) = 0$, then $r$ is indeed a zero of the polynomial.
The zeros of the polynomial can be found using synthetic division, and if the division yields a remainder of zero, that root $r$ is a zero of the polynomial.
More Information
Synthetic division is useful because it simplifies the process of dividing polynomials, especially when you need to find roots. It is much quicker than traditional polynomial long division.
Tips
- Forgetting to bring down the first coefficient correctly.
- Mixing up the operations (addition and multiplication).
- Not checking the final result by substituting back into the original polynomial.
