A polynomial in X of third degree which will vanishes when X equal to 1 and X equal to minus 2 and will have the values 4 and 28 when X equal to minus 1 and X equal to 2 respectively is......

To find a third-degree polynomial that meets the specified conditions, we can start by noting what we know about the polynomial's roots and its behavior at given points. A third-degree polynomial can generally be expressed in the form:

Formulating the Polynomial

Let's denote the polynomial as P(x). Since it vanishes at x = 1 and x = -2, we can express it as:

P(x) = k(x - 1)(x + 2)(x - r)

Here, k is a constant that scales the polynomial, and r is another root we need to determine. The polynomial will be of the form of a cubic equation, which means it has three roots.

Using Known Values

We are given that P(-1) = 4 and P(2) = 28. We can use these points to find the values of k and r. Let's first substitute x = -1 into the polynomial:

P(-1) = k(-1 - 1)(-1 + 2)(-1 - r) = k(-2)(1)(-1 - r) = 2k(1 + r)

Setting this equal to 4 gives us:

• 2k(1 + r) = 4

From this, we simplify to find:

• k(1 + r) = 2 (Equation 1)

Now, let's substitute x = 2 into the polynomial:

P(2) = k(2 - 1)(2 + 2)(2 - r) = k(1)(4)(2 - r) = 4k(2 - r)

Setting this equal to 28 gives us:

• 4k(2 - r) = 28

From this, we simplify to find:

• k(2 - r) = 7 (Equation 2)

Solving the System of Equations

Now we have a system of two equations:

• k(1 + r) = 2 (Equation 1)
• k(2 - r) = 7 (Equation 2)

Let's solve for k in terms of r using Equation 1:

• k = 2 / (1 + r)

Substituting k into Equation 2 gives us:

(2 / (1 + r))(2 - r) = 7

Cross-multiplying leads to:

2(2 - r) = 7(1 + r)

This simplifies to:

• 4 - 2r = 7 + 7r

Combining like terms results in:

• 4 - 7 = 2r + 7r
• -3 = 9r
• r = -1/3

Finding the Value of k

Now that we have r, we can substitute r back into Equation 1 to find k:

k(1 - 1/3) = 2

k(2/3) = 2

Thus, k = 3.

Formulating the Final Polynomial

Now we can plug k and r back into our polynomial expression:

P(x) = 3(x - 1)(x + 2)(x + 1/3).

To simplify this, we can expand it:

P(x) = 3[(x - 1)(x + 2)](x + 1/3).

First, let's expand (x - 1)(x + 2):

• (x - 1)(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2.

Now we multiply this by (x + 1/3):

P(x) = 3(x^2 + x - 2)(x + 1/3) = 3[x^3 + (1 + 1/3)x^2 - 2x - 2/3].

Finally, distributing the 3 gives us the complete polynomial:

P(x) = 3x^3 + 4x^2 - 6x - 2.

Final Result

Therefore, the polynomial you are looking for is:

P(x) = 3x^3 + 4x^2 - 6x - 2.