Question
Which of the following quartic functions has x = –2 and x = –3 as its only two real zeros?
Responses
answer is x4 + 5x3 + 7x2 + 5x + 6
show work
Responses
answer is x4 + 5x3 + 7x2 + 5x + 6
show work
Answers
Answered by
GPT 3.5
To find the quartic function with x = -2 and x = -3 as its only two real zeros, we need to construct the equation using these given zeros.
If x = -2 is a zero, then (x + 2) must be a factor of the equation.
If x = -3 is a zero, then (x + 3) must also be a factor of the equation.
Multiplying (x + 2) and (x + 3) together, we get:
(x + 2)(x + 3) = x² + 5x + 6
This quadratic expression is the product of the factors (x + 2) and (x + 3).
Now, to express it as a quartic function, we multiply it by another quadratic factor. Since it is a quartic function, we need to multiply by x² to make it a degree four polynomial.
(x + 2)(x + 3) * x² = (x² + 5x + 6) * x²
Multiplying the expression (x² + 5x + 6) by x², we get:
x⁴ + 5x³ + 6x²
Therefore, the quartic function with x = -2 and x = -3 as its only real zeros is:
f(x) = x⁴ + 5x³ + 6x²
If x = -2 is a zero, then (x + 2) must be a factor of the equation.
If x = -3 is a zero, then (x + 3) must also be a factor of the equation.
Multiplying (x + 2) and (x + 3) together, we get:
(x + 2)(x + 3) = x² + 5x + 6
This quadratic expression is the product of the factors (x + 2) and (x + 3).
Now, to express it as a quartic function, we multiply it by another quadratic factor. Since it is a quartic function, we need to multiply by x² to make it a degree four polynomial.
(x + 2)(x + 3) * x² = (x² + 5x + 6) * x²
Multiplying the expression (x² + 5x + 6) by x², we get:
x⁴ + 5x³ + 6x²
Therefore, the quartic function with x = -2 and x = -3 as its only real zeros is:
f(x) = x⁴ + 5x³ + 6x²