Problem 44 - Algebra and Trigonometry

Question

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=2 x^{3}-21 x^{2}+9 x-200, \quad c=11 $$

Step-by-Step Solution

Answer

The value of P(11) is 20.

Step 1: Set Up Synthetic Division

Write down the coefficients of the polynomial: 2, -21, 9, and -200. Write the constant term c = 11 at the left of the division symbol.

Step 2: Bring Down the First Coefficient

Bring down the first coefficient, which is 2, to the bottom row as the first entry without any modifications.

Step 3: Multiply and Add

Multiply the brought down number (2) by the constant c (11) and write the result (22) under the next coefficient (-21). Add -21 and 22 to get 1, then write this sum below.

Step 4: Repeat Multiply and Add

Multiply the sum (1) by 11, which is 11. Write this result under the next coefficient (9). Add 9 and 11 to get 20, and write this below the line.

Step 5: Final Multiply and Add Step

Multiply the last sum (20) by 11, resulting in 220. Add this to the final coefficient (-200) to get 20. Write this sum under the line as the remainder.

Step 6: Interpret the Remainder

According to the Remainder Theorem, the remainder 20 is the value of P(11).

Key Concepts

Remainder Theorem

The Remainder Theorem is an essential concept when dealing with polynomials, especially in the context of division. This theorem states that if a polynomial \( P(x) \) is divided by a linear divisor \( x - c \), the remainder of this division is \( P(c) \). Essentially, it provides a shortcut to evaluate the polynomial at a specific value of \( x \). Instead of performing direct multiplication, one can perform division using simpler steps, hence saving time.
In our original exercise, we used the Remainder Theorem to find \( P(11) \). After going through the synthetic division process, we found the remainder to be 20, meaning \( P(11) = 20 \). This illustrates the theorem's power in simplifying complex calculations.

Polynomial Evaluation

Evaluating a polynomial is the process of finding the output value once the input value is given. This task can seem daunting for polynomials of higher degrees due to the number of terms and computations needed. However, with the Remainder Theorem and synthetic division, the process becomes more straightforward.
In our specific problem, we needed to evaluate \( P(11) \) for the given polynomial \( P(x) = 2x^3 - 21x^2 + 9x - 200 \). By applying synthetic division, we carefully reduced the polynomial into smaller parts and added up the results. This allowed us to accurately obtain the result quickly and with fewer operations than traditional methods. The final remainder at the end of the division process directly gave us the value of the polynomial for the specified \( c \).
Through this method, polynomial evaluation becomes not only accurate but also efficient, especially for high-degree polynomials.

Cubic Polynomial

A cubic polynomial is one of the simplest forms of polynomials consisting of four terms: generally, a term in \( x^3 \), \( x^2 \), \( x \), and a constant. The notation of a cubic polynomial typically looks like \( ax^3 + bx^2 + cx + d \), and it plays a crucial role in many mathematical computations.
In our exercise, the polynomial in question was \( 2x^3 - 21x^2 + 9x - 200 \). By recognizing its form, you can apply methods such as synthetic division to evaluate or simplify it more effectively. Due to the degree of the polynomial, the highest exponent determines its name 'cubic' and also suggests that its graph would intercept the x-axis at most three times.
Understanding cubic polynomials is vital, as they frequently appear in various applications, including physics, engineering, and computational mathematics. They allow for an exploration of real-world problems where direct intersections, rates of change, or volumes are calculated.