Polynomial and Intercepts

To find the value of ab bc ca, we need to analyze the characteristics of the polynomial that intersects the x-axis at 1 and -1 and the y-axis at 2. Let's break down the problem step by step.

Intercepts
- The polynomial intersects the x-axis at 1 and -1. This means that when we substitute x = 1 and x = -1 into the polynomial, the result will be zero.
- Let's consider the polynomial in factored form: a(x - 1)(x + 1)(x - k), where k is another root of the polynomial.

Y-Intercept
- The polynomial intersects the y-axis at 2. When we substitute x = 0 into the polynomial, the result should be 2.
- The factored form can be written as: a(0 - 1)(0 + 1)(0 - k) = 2.
- Simplifying, we have -a(1)(-k) = 2.
- This equation gives us ak = 2.

Substituting x = 1 and x = -1
- Let's substitute x = 1 into the factored form: a(1 - 1)(1 + 1)(1 - k) = 0.
- Simplifying, we have a(0)(2)(1 - k) = 0.
- It follows that k = 1.

- Now, let's substitute x = -1 into the factored form: a(-1 - 1)(-1 + 1)(-1 - k) = 0.
- Simplifying, we get a(-2)(0)(-1 - k) = 0.
- This equation gives us k = -1.

Value of ab bc ca
- We have found that k = 1 and k = -1. Since these are the only values of k, we can conclude that k = 1 and k = -1 simultaneously.
- Substituting these values into the equation ak = 2, we have a(1) = 2 and a(-1) = 2.
- This implies that a = 2 and a = -2.
- Therefore, the values of ab bc ca are ab = 2 * 2 = 4, bc = -2 * 2 = -4, and ca = 2 * -2 = -4.

Conclusion
- The values of ab bc ca are 4, -4, -4 respectively, based on the polynomial that intersects the x-axis at 1 and -1 and the y-axis at 2.