# Solving a Third-Degree Polynomial with Real Coefficients

• MHB
• anemone
In summary, a third-degree polynomial is a polynomial with a term of degree three, also known as a cubic polynomial. It can be solved using methods such as factoring, the rational root theorem, or the cubic formula. These solutions can involve complex numbers, but if the polynomial has all real coefficients, the complex solutions will come in pairs. The discriminant, b^2-4ac, can also determine if the polynomial has no real solutions. Solving third-degree polynomials with real coefficients is important in various fields of mathematics as it allows for the determination of the polynomial's roots and understanding its behavior.
anemone
Gold Member
MHB
POTW Director
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 $|f(0)|$.

Because a cubic polynomial shall have maximum 2 humps ( that is where it goes from down to up and from
up to down ) we shall have $f(1) = f(5)= f(6)$ and $f(2) = f(3) = f(7)$ and $f(1) \ne f(2)$.

For them to have values not meeting this criteria we shall have more humps .

As we are interested in absolute value we can choose $f(1) = - f(2) = 12$

As $f(1) = f(5)= f(6) = 12$ So we have $f(1) - 12 = f(5) - 12 = f(6) - 12 = 0$

So we can choose $f(x) = A(x-1)(x-5)(x-6) + 12$ where A is a constant whose value we need to find

Putting $x=2$ we get $f(2) = A* 1 * (-3) *(-4) + 12 = 12A + 12 = - 12$ or A = -2

So $f(x) = -2(x-1)(x-5)(x-6) + 12$

Or $f(0) = -2 * (-1)(-5)(-6) + 12 = 72$

Or $\left | f(0) \right |= 72$

## 1. What is a third-degree polynomial?

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.

## 2. How do you solve a third-degree polynomial with 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.

## 3. What is the Rational Root Theorem?

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.

## 4. What is synthetic division?

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.

## 5. Can a third-degree polynomial have complex roots?

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.

• General Math
Replies
3
Views
1K
• General Math
Replies
5
Views
1K
• General Math
Replies
7
Views
1K
• General Math
Replies
1
Views
889
• General Math
Replies
1
Views
832
• General Math
Replies
1
Views
759
• General Math
Replies
1
Views
1K
• General Math
Replies
3
Views
1K
• General Math
Replies
1
Views
825
• General Math
Replies
1
Views
880