Question

P is a degree of 3. 6 is a root with multiplicity 1, of 9. 1 is a root, with multiplicity 1, of p. 3 is a root, with multiplicity 1, of p. p(-3)=1080. determine an nth degree polynomial that satisfies the aforementioned conditions.

Answer 1

To find the cubic polynomial that satisfies the given conditions, we set up the polynomial with the factors corresponding to the given roots and solve for the leading coefficient using the given value of the polynomial at x = -3. The resulting polynomial is p(x) = 5x^3 - 50x^2 + 135x - 90.

Step-by-step explanation:

The student's question revolves around finding a cubic polynomial p that fits the given conditions. The conditions are that the polynomial has roots of 6 (with multiplicity 1), 1 (with multiplicity 1), and 3 (with multiplicity 1), and that p(-3) = 1080. Based on these conditions, the polynomial can be formed using the root-factor form:

Let's denote the polynomial as p(x). Since 6, 1, and 3 are roots of the polynomial, with multiplicities of 1, the factors of p(x) would be (x-6), (x-1), and (x-3) respectively. Creating the polynomial using these factors gives us:

p(x) = a(x - 6)(x - 1)(x - 3)

Where a is the leading coefficient. We can use the condition p(-3) = 1080 to solve for a:

p(-3) = a((-3) - 6)((-3) - 1)((-3) - 3) = 1080
p(-3) = a(-9)(-4)(-6) = 1080
a = 1080 / 216 = 5

Therefore, the polynomial is:

p(x) = 5(x - 6)(x - 1)(x - 3)

And expanded:

p(x) = 5x^3 - 50x^2 + 135x - 90