I understand how to do these kind of problems except this one.
Write the equation that has the given roots.
Roots: 3 with multiplicity of 2, -5, 0 with multiplicity of 4.
By the way I understood the help that I got earlier, but this problem seems to be more confusing.
(x-3)(x-3)(x+5)x^4 = 0
To write an equation with the given roots, we need to use the concept of "factored form" of a polynomial. The factored form of a polynomial is a multiplication of linear factors, where each factor corresponds to a root of the polynomial.
In this case, we have four roots: 3 with a multiplicity of 2, -5, and 0 with a multiplicity of 4.
To get started, let's write the linear factors corresponding to each root:
(𝑥 − 3)(𝑥 − 3) = (𝑥 − 3)² (for the root 3 with multiplicity 2)
(𝑥 + 5) (for the root -5)
𝑥 (𝑥) (𝑥) (𝑥) = 𝑥⁴ (for the root 0 with multiplicity 4)
Now, we can write the equation by multiplying all the linear factors together:
𝑦 = (𝑥 − 3)² (𝑥 + 5) 𝑥⁴
Simplifying further, we can expand the squared term using the FOIL method:
𝑦 = (𝑥² − 6𝑥 + 9) (𝑥 + 5) 𝑥⁴
Expanding this further, we get:
𝑦 = 𝑥⁷ + 5𝑥⁶ − 6𝑥⁵ + 9𝑥⁵ − 30𝑥⁴ + 45𝑥⁴
Combining like terms:
𝑦 = 𝑥⁷ + 5𝑥⁶ + 3𝑥⁵ + 15𝑥⁴
So, the equation that has the given roots is 𝑦 = 𝑥⁷ + 5𝑥⁶ + 3𝑥⁵ + 15𝑥⁴.