$f(x)=8-4x$ has one solution.
$f(x)=x^2-9$ has two solutions.
$f(x)=x^3+3x^2+5x+15$ has three solutions.
Make a Conjecture:
A polynomial of degree $n$ appears to have $\square$ solutions.
Answer (1)
Observe that a polynomial of degree 1 has 1 solution, a polynomial of degree 2 has 2 solutions, and a polynomial of degree 3 has 3 solutions.
Conjecture that the number of solutions of a polynomial is equal to its degree.
Therefore, a polynomial of degree n has n solutions.
The final answer is n .
Explanation
Understanding the Problem We are given three examples of polynomials and the number of solutions they have. We need to make a conjecture about the number of solutions a polynomial of degree n has.
Analyzing the Examples The first polynomial, f ( x ) = 8 − 4 x , has degree 1 and one solution. The second polynomial, f ( x ) = x 2 − 9 , has degree 2 and two solutions. The third polynomial, f ( x ) = x 3 + 3 x 2 + 5 x + 15 , has degree 3 and three solutions.
Making the Conjecture Based on these examples, we can conjecture that a polynomial of degree n has n solutions.
Conclusion Therefore, a polynomial of degree n appears to have n solutions.
Examples
In electrical engineering, when designing circuits, the behavior of the circuit can often be modeled by polynomial equations. The degree of the polynomial corresponds to the complexity of the circuit, and the solutions of the polynomial represent the possible states or operating points of the circuit. Knowing that a polynomial of degree n has n solutions helps engineers predict and control the behavior of these circuits, ensuring they operate as intended. For example, a cubic polynomial model might represent a circuit with three possible stable states.
