Find a and b if(x+1) (x-1) are the factors of 2x^3-ax^2+bx-5

Given that \((x+1)\) and \((x-1)\) are factors of the polynomial \(2x^3 - ax^2 + bx - 5\), we can use the fact that if \(x - c\) is a factor of a polynomial \(P(x)\), then \(P(c) = 0\).

### Step 1: Apply the Factor Theorem

Since \((x+1)\) is a factor, substituting \(x = -1\) into the polynomial should give us 0:

\[
P(-1) = 2(-1)^3 - a(-1)^2 + b(-1) - 5 = 0
\]

Simplify this:

\[
-2 - a - b - 5 = 0
\]

\[
- a - b = 7 \implies a + b = -7 \quad (1)
\]

Similarly, since \((x-1)\) is a factor, substituting \(x = 1\) should also give us 0:

\[
P(1) = 2(1)^3 - a(1)^2 + b(1) - 5 = 0
\]

Simplify this:

\[
2 - a + b - 5 = 0
\]

\[
-a + b = 3 \quad (2)
\]

### Step 2: Solve the System of Equations

We now have a system of two equations:

1. \(a + b = -7\)
2. \(-a + b = 3\)

Add these two equations together:

\[
(a + b) + (-a + b) = -7 + 3
\]

\[
2b = -4 \implies b = -2
\]

Substitute \(b = -2\) into equation (1):

\[
a + (-2) = -7
\]

\[
a = -5
\]

### Step 3: Conclusion

The values of \(a\) and \(b\) are:

\[
\boxed{a = -5, \; b = -2}
\]

Would you like to know more about polynomial factorization or any other topic? Here are some related questions:

1. How to factor polynomials completely?
2. What is the Factor Theorem, and how is it used?
3. How do you solve a system of linear equations?
4. What are some practical applications of polynomials in real life?
5. How to find the roots of a cubic polynomial?

**Tip:** Always double-check your substitution and simplification steps to avoid errors when solving equations!

Find a and b if (x+1)(x-1) are the factors of 2x^3 - ax^2 + bx - 5

Learn how to solve for a and b using the Factor Theorem for the polynomial 2x^3 - ax^2 + bx - 5, given factors (x+1) and (x-1). This problem involves polynomial factorization, systems of equations, and the Factor Theorem. Suitable for students in Grades 10-12.

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