Verified Answer :
- For the division \((x^2 + x + 8) \div (x - 4)\), perform polynomial long division to find the quotient \(x + 5\) and the remainder \(28\).
- For the polynomial \(-x^4 + 5x^2 + kx - 8\), apply the Remainder Theorem by evaluating the expression at \(x=4\).
- Set the result to zero, resulting in \(-184 + 4k = 0\).
- Solve for the constant to find \(k = 46\).
- Final answers: \(\text{Quotient: } x+5, \text{ Remainder: } 28, k = \boxed{46}\).
Explanation
- Solving the Polynomial Division (16)
To solve the division \((x^2 + x + 8) \div (x - 4)\), we can use polynomial long division.
We divide the leading term \(x^2\) by \(x\) to get \(x\).
Next, multiply \(x(x - 4) = x^2 - 4x\), and subtract this from \(x^2 + x + 8\):
\((x^2 + x + 8) - (x^2 - 4x) = 5x + 8\).
Now, divide the leading term \(5x\) by \(x\) to get \(5\).
Multiply \(5(x - 4) = 5x - 20\), and subtract this from \(5x + 8\):
\((5x + 8) - (5x - 20) = 8 + 20 = 28\).
Therefore, the quotient is \(x + 5\) and the remainder is \(28\).
The result can be written as \(\frac{x^2 + x + 8}{x - 4} = x + 5 + \frac{28}{x - 4}\).
- Finding the value of k (17)
To find the value of \(k\) such that the division of \(p(x) = -x^4 + 5x^2 + kx - 8\) by \((x - 4)\) has a remainder of 0, we apply the Remainder Theorem.
The Remainder Theorem states that the remainder of a polynomial \(p(x)\) divided by \((x - c)\) is \(p(c)\).
Here, we want the remainder when dividing by \((x - 4)\) to be 0, so we set \(p(4) = 0\):
\(p(4) = -(4)^4 + 5(4)^2 + k(4) - 8 = 0\).
Calculating the powers:
\(-(256) + 5(16) + 4k - 8 = 0\).
\(-256 + 80 + 4k - 8 = 0\).
Combining the constant terms:
\(-184 + 4k = 0\).
Adding 184 to both sides gives \(4k = 184\).
Dividing by 4 gives \(k = 46\).
- Conclusion
The division in problem 16 results in a quotient of \(x + 5\) and a remainder of \(28\).
The value of \(k\) that makes the division in problem 17 have a remainder of 0 is \(\boxed{46}\).
Examples
Polynomial division is widely used in engineering and physics, for example, when simplifying transfer functions in control systems or analyzing signal processing filters. By dividing complex polynomials, engineers can easily find the poles and zeros of a system, which are crucial for determining stability and performance. Understanding these algebraic techniques allows for efficient manipulation of mathematical models used in various technological fields.