Final answer:
The polynomial of degree 4 with zeros -3, 0, 3, and 5 is obtained by multiplying the binomials of each zero and simplifying the result. The final polynomial is x⁴ - 5x³ - 9x².
Step-by-step explanation:
To find a polynomial of degree 4 that has the given zeros -3, 0, 3, and 5, we start by writing each zero in the form of a binomial. For example, the zero -3 gives us (x + 3), the zero 0 gives us x, the zero 3 gives us (x - 3), and the zero 5 gives us (x - 5). We then multiply these binomials together to get the polynomial.
So, we have:
- (x + 3) for zero -3,
- x for zero 0,
- (x - 3) for zero 3, and
- (x - 5) for zero 5.
Multiplying these together:
(x + 3) × x × (x - 3) × (x - 5)
We do the multiplication step by step. First, let's multiply (x + 3) by (x - 3) to get a difference of squares:
(x + 3)(x - 3) = x2 - 9
Now let's multiply x by (x - 5):
x(x - 5) = x² - 5x
Next, we multiply the two results together:
(x² - 9) × (x²- 5x) = x⁴- 5x³ - 9x²+ 45x
However, since one of the zeros is 0, the x-term will not appear in the final polynomial, simplifying our result to:
x⁴ - 5x³ - 9x²
This is the polynomial of degree 4 with the given zeros.