Polynomial has a degree 3 a root of multiplicity

Delaney Williamson

Delaney Williamson

Answered question

2022-09-15

Polynomial has a degree 3 

a root of multiplicity 2 at x= 2

a root of multiplicity 1 at x=-1

y- intercept of (0,-8)

Answer & Explanation

nick1337

nick1337

Expert2023-06-17Added 777 answers

To find a polynomial with the given specifications, we'll use the information about the roots and the y-intercept.
The polynomial has a degree 3, and the roots are as follows:
- A root of multiplicity 2 at x = 2
- A root of multiplicity 1 at x = -1
We'll start by writing the linear factors corresponding to each root.
For the root at x = 2 with multiplicity 2, we have the linear factor:
(x2)(x2)
For the root at x = -1 with multiplicity 1, we have the linear factor:
(x+1)
To find the y-intercept, we use the point (0, -8). Since the y-intercept occurs when x = 0, we have the linear factor:
(x0)
Now, we can multiply these factors together to obtain the polynomial:
P(x)=(x2)(x2)(x+1)(x0)
To simplify further, we can expand the expression using the distributive property:
P(x)=(x24x+4)(x+1)(x)
Next, we multiply the binomials using the distributive property:
P(x)=(x34x2+4x+x24x+4)(x)
Combining like terms:
P(x)=(x33x2+4)(x)
Finally, we can multiply the remaining binomials:
P(x)=x43x3+4x
Therefore, the polynomial of degree 3 with the given specifications is:
P(x)=x43x3+4x

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