To find the zeroes of a cubic polynomial, you can use the factor theorem and synthetic division.
1. **Guess and Check**: Start by guessing a possible root (zero) of the polynomial. You can use factors of the constant term divided by factors of the leading coefficient as initial guesses.
2. **Synthetic Division**: Perform synthetic division using the guessed root. If the remainder is zero, then the guessed root is a zero of the polynomial.
3. **Factor Theorem**: If synthetic division gives a remainder of zero, it means that \( (x - \text{guessed root}) \) is a factor of the polynomial.
4. **Repeat or Factor Further**: Once you find one factor, you can use the resulting quotient as a new quadratic polynomial and repeat the process to find more roots.
Remember, the sum of the degrees of the factors should add up to the degree of the original polynomial. Continue this process until you have factored the cubic completely.
Note: This method might involve some trial and error, but it's a systematic approach to finding the zeroes of a cubic polynomial.
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Answer:
To find the zeroes of a cubic polynomial, you can use the factor theorem and synthetic division.
1. **Guess and Check**: Start by guessing a possible root (zero) of the polynomial. You can use factors of the constant term divided by factors of the leading coefficient as initial guesses.
2. **Synthetic Division**: Perform synthetic division using the guessed root. If the remainder is zero, then the guessed root is a zero of the polynomial.
3. **Factor Theorem**: If synthetic division gives a remainder of zero, it means that \( (x - \text{guessed root}) \) is a factor of the polynomial.
4. **Repeat or Factor Further**: Once you find one factor, you can use the resulting quotient as a new quadratic polynomial and repeat the process to find more roots.
Remember, the sum of the degrees of the factors should add up to the degree of the original polynomial. Continue this process until you have factored the cubic completely.
Note: This method might involve some trial and error, but it's a systematic approach to finding the zeroes of a cubic polynomial.