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anemone
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Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12$.
Find $|f(0)|$.
Find $|f(0)|$.
A third-degree polynomial is a mathematical expression that contains a variable raised to the third power (cubic term) and may also include lower degree terms. It is written in the form ax^3 + bx^2 + cx + d, where a, b, c, and d are real coefficients.
To solve a third-degree polynomial with real coefficients, you can use the Rational Root Theorem to find possible rational roots, and then use synthetic division and the Remainder Theorem to find the actual roots. Alternatively, you can use the cubic formula, which is a general formula for finding the roots of any third-degree polynomial.
The Rational Root Theorem states that if a polynomial has integer coefficients, then any rational root (in the form of p/q, where p and q are integers) must have a numerator that is a factor of the constant term and a denominator that is a factor of the leading coefficient. This theorem is useful for finding possible rational roots of a polynomial.
Synthetic division is a shortcut method for dividing a polynomial by a linear factor (x - r), where r is a root of the polynomial. It involves writing the coefficients of the polynomial in a specific pattern and performing simple arithmetic operations to find the quotient and remainder. This method is often used in combination with the Rational Root Theorem to find the actual roots of a polynomial.
Yes, a third-degree polynomial can have complex roots. This means that the roots are not real numbers, but instead involve the imaginary unit i (sqrt(-1)). The cubic formula can be used to find the complex roots of a third-degree polynomial with real coefficients.