Question

The polynomial of degree 4,

P(x) has a root of multiplicity 2 at x=3 and roots of multiplicity 1 at x=0 and x = - 4. It goes through the point (5,144). Find a formula for P(x).

Answer 1

The polynomial P(x) can be represented as P(x) = a × (x - 3)² × x × (x + 4), where a is the leading coefficient. To find a, we use the point (5,144), leading to a = 0.8. Thus, the final polynomial is P(x) = 0.8 × (x - 3)² × x × (x + 4).

Step-by-step explanation:

The polynomial P(x) of degree 4 with specified roots can be expressed by its factors based on the given multiplicities of the roots. Since the root at x=3 has multiplicity 2, the factor would be (x - 3)². For the single roots at x=0 and x=-4, the factors are x and (x + 4), respectively. Thus, the polynomial can be written as P(x) = a × (x - 3)² × x × (x + 4), where a is a leading coefficient that needs to be determined.

To find the value of a, we use the fact that the polynomial goes through the point (5,144). Substituting x=5 into the polynomial gives us P(5) = a × (5 - 3)² × 5 × (5 + 4) which simplifies to 144 = a × 4 × 5 × 9. Solving for a, we get a = 144 / (4 × 5 × 9), or a = 0.8. Therefore, the formula for P(x) is P(x) = 0.8 × (x - 3)² × x × (x + 4).

Answer 2

Step-by-step explanation:

Since the polynomial has a root of multiplicity 2 at x = 3, it can be written as (x - 3)^2 times another polynomial Q(x). Also, since it has roots of multiplicity 1 at x = 0 and x = -4, it can be written as (x - 3)^2 * (x - 0) * (x + 4) * Q(x).

Next, we can use the fact that the polynomial goes through the point (5, 144) to find the formula for Q(x). The polynomial P(x) must satisfy P(5) = 144, so we can write:

(5 - 3)^2 * (5 - 0) * (5 + 4) * Q(5) = 144

4 * 9 * 9 * Q(5) = 144

36 * Q(5) = 144

Q(5) = 4

So, the polynomial P(x) can be written as:

P(x) = (x - 3)^2 * (x - 0) * (x + 4) * 4

= 4 * (x - 3)^2 * (x - 0) * (x + 4)

= 4 * (x^2 - 6x + 9) * x * (x + 4)

= 4x^4 - 84x^3 + 378x^2 - 648x + 576

This is the formula for P(x).