Find a polynomial function of degree 3 with the given numbers as zeros.
-5, sqrt3, -sqrt3
The general cubic with given roots r1,r2,r3 is given by:
y=k(x-r1)(x-r2)(x-r3)
where k is a constant.
I will leave it to you to complete the answer. You are welcome to post your answer for verification.
f(-5)= 1^3 +5(1)^2-3(1)-15 = -12.
f(sqrt3) = 1^3 +5(1)^2-3(1)-15 = -12
f(sqrt-3) = 1^3 +5(1)^2-3(1)-15 = -12
Are these correct?
Not quite. However, it was good that you verify the value of f(x) at the zeroes.
Let the roots r1=-5, r2=√3, r3=-√3.
The function should be equal to zero when the roots are substituted for x, i.e. f(-5)=0, f(sqrt3)=0 and f(-sqrt3)=0.
The desired function is:
f(x)=k(x-r1)(x-r2)(x-r3)
=k(x-(-5))(x-√3)(x--√3)
Now check that f(r1), f(r2) and f(r3) all equal zero.
What is r2 and r3 stand for? Not too sure how to check this. Can you show me please I got 10 more problems like this. Thanks!
Let the roots r1=-5, r2=√3, r3=-√3.
Sorry, problem of encoding.
Let the roots r1=-5, r2=√3, r3=-√3.
These are the roots given in the question.
Ok so the function is
f(x)=k(x-r1)(x-r2)(x-r3)
=k(x-(-5))(x-�ã3)(x--�ã3)
Now to check this?
You will need to multiply the terms out to get the polynomial your teacher needs.
The last two terms should give (x²-3) and the whole function is thus:
f(x)= kx³ + 5kx² -3kx -15k
In fact, since the question requires "Find a polynomial...", you could leave out the factor k to get
f(x)=x³+5x²-3x-15
To find a polynomial function of degree 3 with the given zeros, you can use the factored form of a polynomial. The factored form is given by:
f(x) = a(x - r1)(x - r2)(x - r3)
where a is a constant, and r1, r2, and r3 are the zeros of the polynomial.
In this case, the zeros are -5, √3, and -√3. Let's plug them into the factored form:
f(x) = a(x - (-5))(x - √3)(x - (-√3))
Simplifying this expression, we have:
f(x) = a(x + 5)(x - √3)(x + √3)
Now, to find the value of a, we need additional information. If you know a point on the graph of the polynomial, we can substitute the x and y coordinates of that point into the expression to solve for a. Without any additional information, we cannot determine the exact value of a.
Therefore, a polynomial function of degree 3 with the given zeros -5, √3, and -√3 is:
f(x) = a(x + 5)(x - √3)(x + √3)