If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is
Question
If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is
Solution
The given function f(x) is an even function because f(5 + x) = f(5 - x) for every real x. This means that the function is symmetric about the vertical line x = 5.
Given that f(x) = 0 has four distinct real roots, let's say the roots are a, b, c, and d. Because of the symmetry, two of these roots will be on one side of the line x = 5 and the other two will be on the other side. Let's say a and b are on the left side (x < 5) and c and d are on the right side (x > 5).
Because of the symmetry, a and b are equidistant from the line x = 5, and so are c and d. This means that a + b = c + d.
The sum of the roots is a + b + c + d. Substituting a + b = c + d, we get 2(a + b) = a + b + c + d. Therefore, the sum of the roots is 2(a + b) = 2 * 5 = 10.
Similar Questions
Which of the following polynomial functions have exactly 5 roots, including all real and imaginary roots?Responsesf(x)=5x4+x3−x2+x−5𝑓(𝑥)=5𝑥4+𝑥3−𝑥2+𝑥−5f of x is equal to 5 x to the 4th power plus x cubed minus x squared plus x minus 5f(x)=4x5+3x4+2x3+x2+x𝑓(𝑥)=4𝑥5+3𝑥4+2𝑥3+𝑥2+𝑥f of x is equal to 4 x to the 5th power plus 3 x to the 4th power plus 2 x raised to the 3 power plus x squared plus xf(x)=5x6−5x4+5x2−5𝑓(𝑥)=5𝑥6−5𝑥4+5𝑥2−5f of x is equal to 5 x to the 6th power minus 5 x to the 4th power plus 5 x squared minus 5f(x)=4x4−3x3+2x2−1
Show that 3x + 4x = 5x has exactly one real root
If f(x) is a 4th degree polynomial, and has 3 real numbers as roots, the other root is?Question 4Select one:a.0b.Realc.Complexd.1e.None of these
Youif x to the power 4 -500=0, then sum of roots of equation is ?
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.x4 + x − 5 = 0, (1, 2)f(x) = x4 + x − 5 is on the closed interval [1, 2], f(1) = , and f(2) = . Since −3 < < 13, there is a number c in (1, 2) such that f(c) = by the Intermediate Value Theorem. Thus, there is a of the equation x4 + x − 5 = 0 in the interval (1, 2).
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.