Whats the right answer
What is a quartic function with only the two real zeroes given?
x = 5 and x = 1
(1 point)
Responses
y = –x4 – 6x3 + 6x2 – 6x + 5
y = – x 4 – 6x 3 + 6x 2 – 6x + 5
y = x4 + 6x3 – 6x2 + 6x – 5
y = x 4 + 6x 3 – 6x 2 + 6x – 5
y = x4 – 6x3 + 5x2 – 6x + 6
y = x 4 – 6x 3 + 5x 2 – 6x + 6
y = x4 – 6x3 + 6x2 – 6x + 5
The correct answer is y = x4 – 6x3 + 6x2 – 6x + 5.
The right answer is:
y = x4 – 6x3 + 6x2 – 6x + 5
To find the quartic function with the given real zeroes, you can use the fact that if a number x is a zero of a polynomial function, then the expression (x - a) will be a factor of the polynomial, where a is the zero.
In this case, the two given real zeroes are x = 5 and x = 1. Therefore, the corresponding factors of the quartic function will be (x - 5) and (x - 1).
To find the quartic function, you need to multiply these factors together. Let's expand it step by step:
(x - 5)(x - 1)
= x(x - 1) - 5(x - 1) (using the distributive property)
= x^2 - x - 5x + 5 (simplifying)
= x^2 - 6x + 5
Now, to find the quartic function, you square this expression because it is a quadratic:
(y = x^2 - 6x + 5)^2
= (x^2 - 6x + 5)(x^2 - 6x + 5)
= x^4 - 6x^3 + 5x^2 - 6x^3 + 36x^2 - 30x + 5x^2 - 30x + 25
= x^4 - 12x^3 + 46x^2 - 60x + 25
Therefore, the correct answer is y = x^4 - 12x^3 + 46x^2 - 60x + 25.