Find a polynominal of degree 4 that has integer coefficients and zeros 4,-4,5 and 1/2.
just multiply the factors:
f(x) = (x-4)(x+4)(x-5)(2x-1)
Why did the polynomial go to therapy? Because it had multiple "zero" issues! But don't worry, I've got just the polynomial for you. Since the zeros are 4, -4, 5, and 1/2, the polynomial will have factors (x-4), (x+4), (x-5), and (2x-1).
Now, let's multiply these factors together to get the polynomial:
(x - 4)(x + 4)(x - 5)(2x - 1)
Just expand and simplify, and voila! You've got your degree 4 polynomial with integer coefficients and all those zeros.
To find a polynomial with integer coefficients and zeros 4, -4, 5, and 1/2, we can use the fact that if a number is a zero of a polynomial, then (x - zero) is a factor of the polynomial.
Given the zeros 4, -4, 5, and 1/2, we can use the factors (x - 4), (x + 4), (x - 5), and (x - 1/2) to construct the polynomial.
The polynomial can be written as:
(x - 4)(x + 4)(x - 5)(x - 1/2)
To simplify this expression, we can expand it using the FOIL method and basic algebra:
(x - 4)(x + 4)(x - 5)(x - 1/2)
= (x^2 - 16)(x - 5)(x - 1/2)
= (x^2 - 16)(x^2 - 6.5x + 5/2)
= x^4 - 6.5x^3 + 5/2x^2 - 16x^2 + 104x - 80 - 13/2x + 10 - 5/4
= x^4 - 6.5x^3 - 10.5x^2 + 90.5x - 75/4
Therefore, the polynomial of degree 4 with integer coefficients and zeros 4, -4, 5, and 1/2 is
f(x) = x^4 - 6.5x^3 - 10.5x^2 + 90.5x - 75/4.
To find a polynomial of degree 4 with integer coefficients and the given zeros, we can make use of the fact that if a polynomial has a zero "a", then (x - a) is a factor of the polynomial.
Given zeros: 4, -4, 5, and 1/2.
Step 1: Determine the factors of the polynomial.
Since the zeros are 4, -4, 5, and 1/2, the factors of the polynomial are:
(x - 4), (x + 4), (x - 5), and (2x - 1).
Step 2: Multiply the factors to get the polynomial.
Multiply the factors together to obtain the desired polynomial:
(x - 4)(x + 4)(x - 5)(2x - 1)
Expanding this expression, we get:
(x^2 - 16)(x - 5)(2x - 1)
Now, multiply the two binomials:
(x^2 - 16)(2x^2 - 11x + 5)
Multiply each term in the first binomial by each term in the second binomial:
2x^4 - 11x^3 + 5x^2 - 32x^2 + 176x - 80
Combining like terms, we simplify the expression:
2x^4 - 11x^3 - 27x^2 + 176x - 80
Therefore, a polynomial of degree 4 with integer coefficients and zeros 4, -4, 5, and 1/2 is:
2x^4 - 11x^3 - 27x^2 + 176x - 80.