Question

Write and graph a polynomial function that has one real zero in each of the intervals \(-2

Short answer

The function \(f(x) = a(x + 1.5)(x - 0.5)(x - 4.5)\) is a polynomial that has one real zero in each of the intervals \(-2<x<-1,0<x<1\), and \(4<x<5\). This is a cubic polynomial of degree 3, but infact, there is no maximum degree for such a function as any polynomial of degree greater than 3 possessing additional pairs of non-real complex conjugate roots could also satisfy this condition.

Step-by-step solution

Building the Polynomial Function

One real zero in each of the intervals means that the function crosses the x-axis at those points. The simplest approach is to make the roots -1.5, 0.5, and 4.5 - each which lies in one of our intervals. A polynomial equation with those roots would look like this: \( f(x) = a(x + 1.5)(x - 0.5)(x - 4.5) \) . A pre-factor 'a' is chosen to allow for any vertical stretching or compression.

Graphing the Function

When plotting the function, make sure to mark the roots on the x-axis and where they cut the x-axis. Also, the polynomial will downward facing if 'a' is negative, as the product of negative 'a' with odds negative roots results in a negative value. The polynomial becomes positive for values of 'x' that are to the right of 4.5 and negative for values of 'x' to the left of -1.5.

Answering the Degree Question

Every polynomial of degree n can have up to n real roots or zeros. In this case, the function has three real roots, so it's a cubic polynomial whose degree is three. However, there isn't a maximum possible degree for such a function. Additional pairs of non-real complex conjugate roots could be added to the function indefinitely, increasing the degree of the function. Simply, any polynomial of degree greater than 3 that possesses additional roots in the form of conjugate pairs which are not real numbers, could also meet this criteria.

Key concepts

Real Roots

Real roots of a polynomial are the x-values where the graph intersects the x-axis. These are crucial points because they indicate the solutions to the polynomial equation when set to zero.
For a polynomial to have real roots in specified intervals like - \(-2 < x < -1\)- \(0 < x < 1\) - \(4 < x < 5\)
The polynomial must cross the x-axis within each of these intervals, ensuring that each has one real-zero within.
To find these, we often solve for roots between particular values based on our established intervals. This interaction between the function and the x-axis helps to give a visual representation of these points.

Graphing Polynomials

Graphing a polynomial function involves visually representing the equation on a coordinate plane. This allows us to understand the behavior of the function, including where real roots are located.
The graph of a polynomial function will have several characteristic features:

• Intersects the x-axis at roots; dip or turn at these points can vary based on whether they are single or repeated roots.
• The end behavior is determined by the leading term which indicates how the graph behaves as it approaches infinity in both directions.
• If we choose 'a' in a polynomial function like \(a(x + 1.5)(x - 0.5)(x - 4.5)\) as negative, the graph will invert and open downward rather than upward.

A detailed graph aids in identifying crucial points, like turning points and intersections, which inform us about polynomial behavior over its domain.

Degree of Polynomial

The degree of a polynomial function provides insight into the number, and potentially the nature, of its roots. Polynomial degree is determined by the highest power of x present.

• A degree of three means the polynomial is cubic, typically allowing for up to three real roots or zeros.
• The degree dictates the polynomial's shape and complexity — the higher, the more potential for twists and turns.
• Higher-degree polynomials can have real and complex roots, establishing its graph's absolute overall shape.

Understanding a polynomial's degree helps us predict its shape and ensure it meets interval conditions by observing how many times it crosses the x-axis.

Complex Roots

Complex roots usually occur when a polynomial's discriminant isn't positive, which results in non-real numbers. Complex roots often appear as conjugate pairs - expressions in the form of \(a + bi\) and \(a - bi\), where \(i\) is the imaginary unit.

• These don't appear on the x-axis like real roots but affect the polynomial's degree.
• Adding complex roots to a cubic polynomial increases its degree while potentially maintaining the count of real roots.
• A polynomial with only real coefficients has complex roots in conjugate pairs to ensure the overall expression remains real.

Complex roots explain why there's no maximum degree for a polynomial meeting certain real root criteria since these roots are traditionally additive without changing the visible real root structure.