The polynomial x^4-x^3-5x^2+ax-4 is denoted by p(x). It is given that p(x) is divisible by x^2-2x-4. Find the value of a

Given that the polynomial \( p(x) = x^4 - x^3 - 5x^2 + ax - 4 \) is divisible by \( x^2 - 2x - 4 \), we can express \( p(x) \) as:

\[
p(x) = (x^2 - 2x - 4)q(x)
\]

where \( q(x) \) is a polynomial of degree 2. Since \( p(x) \) is divisible by \( x^2 - 2x - 4 \), the roots of \( x^2 - 2x - 4 \) must also be the roots of \( p(x) \).

First, find the roots of \( x^2 - 2x - 4 \). Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1 \), \( b = -2 \), and \( c = -4 \):

\[
x = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = \frac{2 \pm 2\sqrt{5}}{2} = 1 \pm \sqrt{5}
\]

Thus, the roots are \( 1 + \sqrt{5} \) and \( 1 - \sqrt{5} \).

Since \( p(x) \) is divisible by \( x^2 - 2x - 4 \), \( p(1 + \sqrt{5}) = 0 \) and \( p(1 - \sqrt{5}) = 0 \).

Substitute \( x = 1 + \sqrt{5} \) into \( p(x) \):

\[
(1 + \sqrt{5})^4 - (1 + \sqrt{5})^3 - 5(1 + \sqrt{5})^2 + a(1 + \sqrt{5}) - 4 = 0
\]

First, calculate each term:

\[
(1 + \sqrt{5})^2 = 1 + 2\sqrt{5} + 5 = 6 + 2\sqrt{5}
\]

\[
(1 + \sqrt{5})^3 = (1 + \sqrt{5})(6 + 2\sqrt{5}) = 6 + 2\sqrt{5} + 6\sqrt{5} + 10 = 16 + 8\sqrt{5}
\]

\[
(1 + \sqrt{5})^4 = (1 + \sqrt{5})(16 + 8\sqrt{5}) = 16 + 8\sqrt{5} + 16\sqrt{5} + 40 = 56 + 24\sqrt{5}
\]

Now substitute these back into the polynomial:

\[
(56 + 24\sqrt{5}) - (16 + 8\sqrt{5}) - 5(6 + 2\sqrt{5}) + a(1 + \sqrt{5}) - 4 = 0
\]

\[
56 + 24\sqrt{5} - 16 - 8\sqrt{5} - 30 - 10\sqrt{5} + a + a\sqrt{5} - 4 = 0
\]

Combine like terms:

\[
(56 - 16 - 30 - 4) + (24\sqrt{5} - 8\sqrt{5} - 10\sqrt{5} + a\sqrt{5}) + a = 0
\]

\[
6 + (6\sqrt{5} + a\sqrt{5}) + a = 0
\]

Separate real and imaginary parts:

\[
6 + a + (6 + a)\sqrt{5} = 0
\]

Since both parts must be zero:

\[
6 + a = 0 \quad \text{and} \quad 6 + a = 0
\]

Thus, \( a = -6 \).

Therefore, the value of \( a \) is \( \boxed{-6} \).

Would you like further details or have any questions?

Here are some questions you might want to ask next:
1. Can you explain how to use the quadratic formula?
2. How do you simplify expressions with square roots?
3. What are some properties of polynomial division?
4. How can you verify if a polynomial is divisible by another polynomial?
5. Can you show another example of polynomial division?

Tip: When working with polynomials, understanding the roots and how to manipulate them can help simplify the problem significantly.

Discover how to find the value of 'a' in a polynomial problem where p(x) is divisible by x^2 - 2x - 4. Explore the step-by-step solution involving polynomials, quadratic equations, and roots. Suitable for high school students studying algebra and polynomial division.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation