Math Problem Statement
solve
Solution
Let's address the questions based on the polynomial function and its characteristics.
Question 12:
The function given is: To determine how many x-intercepts this function has, we need to set and solve for :
Rearranging the equation:
Let . Substituting gives: We solve this quadratic equation using the quadratic formula:
Since is a positive real number, we get two real solutions for . However, , so each solution for will yield two values of (since ).
Both solutions for are positive, indicating real solutions for . Therefore, the function has 4 x-intercepts.
Answer for Question 12: 4
Question 13:
For an even polynomial function with a positive leading coefficient, the minimum number of x-intercepts is 0. This is because the graph of an even-degree polynomial (like ) opens either upwards or downwards. In some cases, it may not intersect the x-axis at all.
Answer for Question 13: 0
Question 14:
For an even polynomial function with a positive leading coefficient, the maximum number of x-intercepts is determined by the degree of the polynomial. Since the highest degree of the polynomial is 4 (even degree), the function can have a maximum of 4 real x-intercepts.
Answer for Question 14: 4
Would you like any further explanations or have more questions? Here are some related questions to explore further:
- How do the x-intercepts of odd-degree polynomials differ from even-degree ones?
- How do you determine the number of turning points for a polynomial function?
- What is the role of the leading coefficient in determining the direction of the graph of a polynomial?
- How do you solve a quadratic equation using the quadratic formula step by step?
- What happens when a polynomial function has repeated roots?
Tip: When solving polynomial equations, always check if factoring can simplify the process before using more complex methods like the quadratic formula.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Polynomial Functions
X-Intercepts
Even Degree Polynomials
Formulas
Quadratic formula: z = (-b ± √(b² - 4ac)) / 2a
Substitution method for solving higher-degree polynomials
Theorems
Fundamental Theorem of Algebra
Behavior of Even Degree Polynomials
Suitable Grade Level
Grades 10-12
Related Recommendation
Analyze Polynomial y = x^4 - x^3 - 2x^2: Intercepts and End Behavior
Odd Degree Polynomial with Intercepts at (-2,0) and (5,0), Y-Intercept (0,-10)
Determine the Form of a Polynomial Function: y = -x^4 - 2x^3 + 3x^2 + 4x - 4
Find the x- and y-intercepts of Polynomial Functions
Intercepts of Polynomial Function f(x) = x^3 + 5x^2 - 9x - 45