A)p(x) = x4 + 4x³ -3x² - 10x + 8 B)p(x) = x³ + 5x² + 2x - 8
C)p(x) = x4 - 4x³ -3x² + 10x + 8
D)p(x) = x³ - 5x² + 2x + 8

The polynomial is:

p(x) = x^4 -4x^3-3x^2 +10x+8p(x)=x

4

−4x

3

−3x

2

+10x+8

Solution:

Given that,

A polynomial, p(x), has a lead coefficient of 1 and exactly three distinct zeros

The zeros are:

x = -1 is a zero of multiplicity two

x = 2 is a zero of multiplicity one

x = 4 is a zero of multiplicity one

Zero of multiplicity means "How many times a particular number is a zero for a given polynomial"

Therefore,

x = -1

x + 1 = 0 , with multiplicity of two

x = 2

x - 2 = 0, with multiplicity one

x = 4

x - 4 = 0, with multiplicity one

The polynomial p(x) is given as:

p(x) is equal to product of zeros

p(x) = (x+1)^2(x-2)(x-4)p(x)=(x+1)

2

(x−2)(x−4)

\begin{gathered}p(x) = (x^2 + 2x + 1)(x-2)(x-4)Muliply\ each\ term\ in\ first\ bracket\ with\ each\ term\ in\ second\ bracketp(x) = (x^3 -2x^2 +2x^2 -4x + x - 2)(x-4)p(x) = (x^4-4x^3 -2x^3+8x^2 +2x^3 -8x^2 -4x^2 + 16x +x^2-4x -2x + 8)Simplify\ the\ abovep(x) = x^4 -4x^3-3x^2+10x +8\end{gathered}

p(x)=(x

2

+2x+1)(x−2)(x−4)

Muliply each term in first bracket with each term in second bracket

p(x)=(x

3

−2x

2

+2x

2

−4x+x−2)(x−4)

p(x)=(x

4

−4x

3

−2x

3

+8x

2

+2x

3

−8x

2

−4x

2

+16x+x

2

−4x−2x+8)

Simplify the above

p(x)=x

4

−4x

3

−3x

2

+10x+8

Thus the polynomial is found

oh

Step-by-step explanation:

step-by-step explanation:

hjkj