AP Precalculus Practice Exam Question 23

NumWorks
23 May 202303:11
EducationalLearning
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TLDRThe video script discusses a fourth-degree polynomial function K(x) with non-zero real constants a and b. It explores the properties of its zeros, highlighting that all have multiplicity one and result in crossing x-intercepts. The script refutes statements about the number of x-intercepts and real solutions, and clarifies that one zero is a complex number with its conjugate also being a zero. The true statement is identified as the complex conjugate pair being zeros of the polynomial, debunking the misconceptions about the graph's behavior and the nature of its solutions.

Takeaways
  • 📚 The polynomial function K(x) is a fourth-degree polynomial with the form K(x) = ax^4 - bx^3 + 15.
  • 🔍 The constants a and b are non-zero real numbers.
  • 🎯 The polynomial has exactly four zeros, all with a multiplicity of one.
  • 🌐 One of the zeros is a complex number, implying the existence of its conjugate as another zero.
  • 📍 An x-intercept of the graph of K is at the point (17.997, 0).
  • ✂️ Since the zeros have a multiplicity of one, all x-intercepts are crossing, not tangent to the x-axis.
  • 🔢 Given that there is one complex zero, there must be one real zero, making a total of two real x-intercepts.
  • 🚫 Statement A is incorrect because the graph does not have three x-intercepts; it has two.
  • 🔄 Statement B is correct as it states the conjugate of the complex zero is also a zero.
  • 🚫 Statement C is incorrect because the equation does not have four real solutions; it has two.
  • 🚫 Statement D is incorrect because the graph is not tangent to the x-axis at x = 17.997; it crosses.
Q & A

  • What is the degree of the polynomial function K(x) mentioned in the script?

    -The degree of the polynomial function K(x) is four, as it is given by K(x) = ax^4 - Bx^3 + fifteen.

  • What is the significance of the zeros of K having a multiplicity of one?

    -A multiplicity of one for the zeros of K means that each zero is a simple root, which implies that the graph of K will cross the x-axis at each of these points, rather than touching or being tangent to it.

  • How many x-intercepts does the graph of K have, according to the script?

    -The graph of K has two x-intercepts. This is because two of the zeros are complex, and complex zeros come in conjugate pairs, which do not contribute to additional x-intercepts.

  • What is the x-intercept of the graph of K mentioned in the script?

    -The x-intercept of the graph of K is at the point (17.997, 0).

  • What does it mean for a zero of K to be a complex number?

    -A zero of K being a complex number means that it is not a real number but a number in the form of a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.

  • Why does the presence of a complex zero imply that its conjugate is also a zero of K?

    -Complex zeros of polynomials with real coefficients always come in conjugate pairs due to the complex conjugate root theorem, which states that non-real roots must occur in pairs that are complex conjugates of each other.

  • What is the relationship between the coefficients 'a' and 'B' in the polynomial K(x)?

    -The coefficients 'a' and 'B' are non-zero real constants that determine the specific shape and position of the graph of the polynomial K(x).

  • Why can't the graph of K be tangent to the x-axis at any of its zeros?

    -The graph of K cannot be tangent to the x-axis at any of its zeros because all zeros have a multiplicity of one, which means the graph will cross the x-axis at these points instead of touching it without crossing.

  • What does statement B in the script assert about the zeros of K?

    -Statement B asserts that the conjugate of the given complex zero is also a zero of K, which is a correct statement based on the complex conjugate root theorem.

  • Which of the statements A, B, C, or D is identified as true in the script?

    -Statement B is identified as true in the script, as it correctly states that the conjugate of a complex zero is also a zero of the polynomial K.

Outlines
00:00
📚 Polynomial Zeros and Graph Characteristics

The script discusses a polynomial function K(x) = ax^4 - bx^3 + 15 with non-zero real constants a and b. It clarifies that the polynomial has four zeros, with one being a complex number, which implies its conjugate is also a zero, and the remaining two are real zeros. The multiplicity of each zero is one, indicating crossing x-intercepts rather than tangents. The script refutes statements about the number of x-intercepts and the nature of the roots, concluding that the true statement is that the conjugate of the given complex zero is also a zero of the polynomial.

Mindmap
Keywords
💡Polynomial function
A polynomial function is a mathematical expression involving a sum of powers in a single variable. In the video, the polynomial function K is defined by K(x) = ax^4 - bx^3 + 15, where a and b are non-zero real constants. This function is central to the video's theme as it sets the stage for discussing the properties of its zeros.
💡Zeros
Zeros of a function are the values of x for which the function equals zero. The video discusses the zeros of the polynomial K, noting that all zeros have a multiplicity of one, meaning each zero is a simple root. This concept is essential for understanding the behavior of the polynomial function at specific points.
💡Multiplicity
Multiplicity refers to the number of times a root appears in a polynomial. In the script, it is stated that each zero of K has a multiplicity of one, indicating that the roots are simple and the function crosses the x-axis at these points without touching it at any other point.
💡X-intercept
An x-intercept is the point where a graph of a function crosses the x-axis, which occurs when the function's value is zero. The video mentions an x-intercept at (17.997, 0), illustrating the real zero of the polynomial function K.
💡Complex number
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The video states that one of the zeros of K is a complex number, which implies the existence of a conjugate zero due to the nature of polynomial equations.
💡Conjugate
In the context of complex numbers, a conjugate is a number with the same real part but an opposite sign for the imaginary part. The video correctly identifies that if one zero of the polynomial is complex, its conjugate must also be a zero, which is a key property of polynomials with real coefficients.
💡Degree of a polynomial
The degree of a polynomial is the highest power of the variable in the polynomial expression. The script specifies that the polynomial K has a degree of four, which means it can have up to four zeros, including complex and real zeros.
💡Real solutions
Real solutions are values of the variable that result in a real number when substituted into the function. The video discusses the real solutions of the equation K(x) = 0, noting that there are only two real solutions because the complex zeros come in conjugate pairs.
💡Tangent
A tangent to a curve is a straight line that touches the curve at a single point without crossing it. The video script refutes the statement that the graph of K is tangent to the x-axis at x = 17.997, correctly stating that the graph crosses the x-axis due to the multiplicity of the zeros.
💡Crossing x-intercepts
Crossing x-intercepts refer to the points where a graph intersects the x-axis. The video emphasizes that all zeros of the polynomial K have a multiplicity of one, meaning the graph crosses the x-axis at these points, which is a key observation for understanding the graph's behavior.
Highlights

The polynomial function K is of degree four with coefficients a and b.

All zeros of K have a multiplicity of one.

One of the zeros is a complex number, implying the existence of its conjugate as another zero.

The polynomial has a real x-intercept at the point (17.997, 0).

Complex zeros come in pairs, indicating two complex zeros for the polynomial.

Due to the multiplicity of one, all x-intercepts are crossing points, not tangents.

The graph of K cannot have three x-intercepts because two are complex conjugates.

Statement A is incorrect as it claims three x-intercepts, which contradicts the complex zeros.

Statement B is identified as true, confirming the conjugate of the complex zero is also a zero.

Statement C is false because the equation cannot have four real solutions given the complex zeros.

Statement D is incorrect as the graph of K is not tangent to the x-axis at x equals 17.997.

The true statement is B, which confirms the conjugate pair of complex zeros.

The analysis of the polynomial's zeros and their properties is crucial for understanding the graph's behavior.

The multiplicity of zeros affects the type of x-intercepts the graph will have.

Understanding the relationship between real and complex zeros is key to solving polynomial equations.

The given information about the polynomial's zeros leads to the conclusion that statement B is correct.

Transcripts

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Polynomial AnalysisComplex ZerosReal SolutionsGraph PropertiesMultiplicity OneX-InterceptsRoot PairsTangency TestMathematics EducationAlgebraic ConceptsGraph Behavior