Question

let
\\ p(x) = 4x^3 + 4x^2 - 5x + 10. \\
use long division to calculate \\( p(1) \\).
\\ p(1) = \\

Explanation:

Step1: Apply the Remainder Theorem

Divide \(p(x)\) by \(x - 1\) to find the remainder.

$$\frac{4x^3 + 4x^2 - 5x + 10}{x - 1}$$

Step2: Divide the first term

Divide \(4x^3\) by \(x\) to get \(4x^2\).

$$4x^2(x - 1) = 4x^3 - 4x^2$$

Step3: Subtract and bring down

Subtract \(4x^3 - 4x^2\) from \(4x^3 + 4x^2\).

$$(4x^3 + 4x^2) - (4x^3 - 4x^2) = 8x^2$$

Step4: Divide the next term

Divide \(8x^2\) by \(x\) to get \(8x\).

$$8x(x - 1) = 8x^2 - 8x$$

Step5: Subtract and bring down

Subtract \(8x^2 - 8x\) from \(8x^2 - 5x\).

$$(8x^2 - 5x) - (8x^2 - 8x) = 3x$$

Step6: Divide the final term

Divide \(3x\) by \(x\) to get \(3\).

$$3(x - 1) = 3x - 3$$

Step7: Find the final remainder

Subtract \(3x - 3\) from \(3x + 10\).

$$(3x + 10) - (3x - 3) = 13$$

Answer:

\(p(1) = 13\)