Question

A circle and a line intersect at most twice. A circle and a parabola intersect at most four times. Deduce that a circle and the graph of a polynomial of degree 3 intersect at most six times. What do you conjecture about a polynomial of degree 4 ? What about a polynomial of degree \(n\) ? Can you explain your conclusions using an algebraic argument?

Short answer

A circle and a cubic polynomial intersect at most six times. For a fourth-degree polynomial, they intersect at most eight times. Generally, a circle and a degree \( n \) polynomial intersect at most \( 2 + n \) times.

Step-by-step solution

- Recall Circle and Line Intersections

A circle and a line can intersect at most twice. This can occur at two distinct points, one point (if the line is tangent), or not at all (if the line does not touch the circle).

- Recall Circle and Parabola Intersections

A circle and a parabola can intersect at most four times. This can occur because the highest degree polynomial equation formed when equating the circle's and parabola's equations is a fourth-degree polynomial.

- Analyze Circle and Cubic Polynomial Intersections

To find the maximum number of intersections for a circle and a cubic polynomial (degree 3), equate the circle's equation \[(x - h)^2 + (y - k)^2 = r^2\]and the cubic polynomial's equation \[f(x) = ax^3 + bx^2 + cx + d.\]Substituting \( y = ax^3 + bx^2 + cx + d \) into the circle's equation results in a single polynomial equation of degree 6. Therefore, it can have at most six real solutions.

- Conjecture for Polynomial of Degree 4

Using a similar approach, equate the circle's equation and a fourth-degree polynomial equation. The resulting polynomial equation will be of degree 8, hence it can have at most eight intersections.

- Generalize Conjecture for Polynomial of Degree n

For a polynomial of degree \( n \), substituting the polynomial into the circle's equation will result in a polynomial equation of degree \( 2 + n \). Thus, the circle and the polynomial graph can intersect at most \( 2 + n \) times.

- Algebraic Argument

The intersection points correspond to the solutions of the polynomial equation obtained by equating the circle's equation to the polynomial's equation. The degree of this polynomial equation determines the maximum number of solutions. For a circle and a polynomial of degree \( n \), this equation will have a maximum degree of \( 2 + n \), thus it can have at most \( 2 + n \) real solutions.

Key concepts

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial \(f(x) = ax^3 + bx^2 + cx + d\), the highest power of x is 3, so this is a third-degree polynomial. Polynomials can have degrees ranging from zero (a constant polynomial) to any positive integer. The degree indicates the polynomial's general shape and potential number of turning points. Understanding the degree of a polynomial helps us predict its characteristics and behavior, especially when finding intersection points with geometric shapes like circles.

Intersection Points

Intersection points are where two graphs meet or cross each other. For instance, the intersection of a circle and a line can occur in two places at most. Similarly, a circle and a parabola (a degree 2 polynomial) can intersect at up to four distinct points. When intersecting a circle with a polynomial of degree 3, we substitute the polynomial equation into the circle’s equation, resulting in a new equation with a higher degree. This new equation's degree tells us how many intersection points can exist. For example, a circle and a cubic polynomial can intersect up to six times, as the resulting equation is of degree 6.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. They can sometimes be simple, like \(x + 2 = 5\), and sometimes more complex, involving polynomials and multiple variables. When analyzing intersections, we often set the algebraic equations of two curves equal to each other. For example, the equation of a circle might be \((x - h)^2 + (y - k)^2 = r^2\) and a polynomial \(y = ax^3 + bx^2 + cx + d\). By equating \(y\) from the polynomial to the squaring part of the circle equation, we create a higher-degree algebraic equation from which we can find the possible intersection points.

Circles

A circle’s equation in Cartesian coordinates is usually in the form \((x - h)^2 + (y - k)^2 = r^2\), where \(h\) and \(k\) are the coordinates of the circle’s center and \(r\) is the radius. This equation describes all points (x, y) that are at a distance \(r\) from the center \( (h, k) \). Understanding the circle’s geometry helps us analyze how it can intersect with other shapes, like lines or polynomials. For instance, when a circle intersects with a polynomial graph, the resulting equation helps us determine the maximum number of intersection points.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, involving the operations of addition, subtraction, multiplication, and non-negative integer exponents. They are represented generally as \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where the largest exponent \ is the degree of the polynomial. Each polynomial degree has different properties. For example, a linear polynomial (degree 1) forms a straight line, while a cubic polynomial (degree 3) forms an S-shaped curve. When polynomials intersect with other shapes, like circles, the degree helps determine the number of possible intersection points. Substituting the polynomial equation into another shape's equation (like a circle), results in a higher-degree equation that predicts these intersections.