Math Problem Statement

solve

The function y = -x^4 + 3x^2 + 5 has _____ x-intercepts. For an even polynomial function with a positive leading coefficient, what is the minimum and maximum number of x-intercepts?

Solution

Let's address the questions based on the polynomial function and its characteristics.

Question 12:

The function given is: y=x4+3x2+5y = -x^4 + 3x^2 + 5 To determine how many x-intercepts this function has, we need to set y=0y = 0 and solve for xx:

0=x4+3x2+50 = -x^4 + 3x^2 + 5

Rearranging the equation: x43x25=0x^4 - 3x^2 - 5 = 0

Let z=x2z = x^2. Substituting gives: z23z5=0z^2 - 3z - 5 = 0 We solve this quadratic equation using the quadratic formula: z=(3)±(3)24(1)(5)2(1)=3±9+202=3±292z = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-5)}}{2(1)} = \frac{3 \pm \sqrt{9 + 20}}{2} = \frac{3 \pm \sqrt{29}}{2}

Since 29\sqrt{29} is a positive real number, we get two real solutions for zz. However, z=x2z = x^2, so each solution for zz will yield two values of xx (since x=±zx = \pm \sqrt{z}).

Both solutions for zz are positive, indicating real solutions for xx. 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 x4x^4) 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:

  1. How do the x-intercepts of odd-degree polynomials differ from even-degree ones?
  2. How do you determine the number of turning points for a polynomial function?
  3. What is the role of the leading coefficient in determining the direction of the graph of a polynomial?
  4. How do you solve a quadratic equation using the quadratic formula step by step?
  5. 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.

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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