Question

Determine a polynomal of degree 3 in general from with data zeros -1(double) and -4 ; p(-2)=6

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Solution:

1. Given zeros and multiplicities:

* Zeros:-1 (double root), -4

2. General form of the polynomial with given zeros:

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

3. Use given point to find the coefficienta :

* Given:p(-2) = 6

* Substitutex = -2 into the polynomial:

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

6 = a(-1)^2(2)

6 = 2a

a = 3

4. The polynomial in general form:

p(x) = 3(x + 1)^2(x + 4)

5. Expand the polynomial to standard form:

p(x) = 3(x^2 + 2x + 1)(x + 4)

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

p(x) = 3(x^3 + 6x^2 + 9x + 4)

p(x) = 3x^3 + 18x^2 + 27x + 12

1. Given zeros and multiplicities:

* Zeros:

2. General form of the polynomial with given zeros:

3. Use given point to find the coefficient

* Given:

* Substitute

4. The polynomial in general form:

5. Expand the polynomial to standard form:

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