Problem 50 - Algebra and Trigonometry

Question

Let $$\begin{aligned} P(x)=6 x^{7}-40 x^{6}+16 x^{5}-200 x^{4} & \\\\-&-60 x^{3}-69 x^{2}+13 x-139 \end{aligned}$$ Calculate \(P(7)\) by (a) using synthetic division and (b) substituting \(x=7\) into the polynomial and evaluating directly.

Step-by-Step Solution

Answer

Using both synthetic division and direct substitution, the result is \(P(7) = 1\).

Step 1: Set up for Synthetic Division

To use synthetic division, we first set it up by writing the value 7, which we will substitute into the polynomial, to the left. The coefficients of the polynomial from highest to lowest degree (including zero for any missing powers) are written in a row: 6, -40, 16, -200, -60, -69, 13, -139.

Step 2: Perform Synthetic Division

1. Bring down the first coefficient, 6.2. Multiply this value by 7 and write it under the next coefficient: \((7 \times 6 = 42)\), write 42 under -40.3. Add -40 and 42, resulting in 2.4. Repeat the multiply and add process for each coefficient: \((2 \times 7 = 14), (16 + 14 = 30), (30 \times 7 = 210), (-200 + 210 = 10), (10 \times 7 = 70), (-60 + 70 = 10), (10 \times 7 = 70), (-69 + 70 = 1), (1 \times 7 = 7), (13 + 7 = 20), (20 \times 7 = 140), (-139 + 140 = 1).\)5. The final remainder is what we obtain for \(P(7)\) using synthetic division, which is 1.

Step 3: Substitute Directly and Simplify

Substitute \(x = 7\) directly into the given polynomial: \[P(7) = 6(7^7) - 40(7^6) + 16(7^5) - 200(7^4) - 60(7^3) - 69(7^2) + 13(7) - 139\]Calculate each power of 7, then compute each product and sum:- \(7^7 = 823543\), \(6 \times 823543 = 4941258\)- \(7^6 = 117649\), \(-40 \times 117649 = -4705960\)- \(7^5 = 16807\), \(16 \times 16807 = 268912\)- \(7^4 = 2401\), \(-200 \times 2401 = -480200\)- \(7^3 = 343\), \(-60 \times 343 = -20580\)- \(7^2 = 49\), \(-69 \times 49 = -3381\)- \(13 \times 7 = 91\)Combine all these values to obtain:\[4941258 - 4705960 + 268912 - 480200 - 20580 - 3381 + 91 - 139 = 1\]Thus, the result is the same, \(P(7) = 1\).

Key Concepts

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial expression for a given value of the variable. In the original exercise, we need to evaluate the polynomial \( P(x) \) at \( x = 7 \). This can be achieved using various methods, such as synthetic division or direct substitution.

To start, let's consider the expression for the polynomial:\[ P(x) = 6x^7 - 40x^6 + 16x^5 - 200x^4 - 60x^3 - 69x^2 + 13x - 139. \]Evaluating \( P(7) \) involves substituting \( x = 7 \) into every term of the polynomial. The terms need to be handled carefully to avoid arithmetic errors, ensuring each component is calculated correctly.

Let's consider how this is done using two methods: synthetic division and direct substitution.

Remainder Theorem

The Remainder Theorem is a powerful tool in polynomial algebra. It states that the remainder of the division of a polynomial \( P(x) \) by a linear divisor \( x - a \) is equal to \( P(a) \). In other words, when you divide \( P(x) \) by \( x - 7 \) using synthetic division, the remainder you get is exactly the value of \( P(7) \).

This theorem is particularly helpful because it allows us to evaluate polynomials quickly without performing the complete polynomial division. Instead of going through the rigorous process of polynomial long division, synthetic division offers a streamlined approach where only simple arithmetic operations are required. Thus, obtaining \( P(7) = 1 \) from the remainder of the synthetic division process confirms the accuracy of our evaluations.

• This technique saves time and simplifies calculations by reducing them to elementary operations.
• Despite its simplicity, it requires discipline in performing the sequence of operations accurately to avoid any errors.

Direct Substitution

Direct substitution is a straightforward approach used to evaluate a polynomial by directly replacing the variable with a specific number, then computing the resulting arithmetic expression. In the exercise, we evaluate \( P(x) \) for \( x = 7 \) by substituting 7 into the polynomial equation and calculating step by step.

After substituting \( x = 7 \) into each term of \( P(x) = 6x^7 - 40x^6 + 16x^5 - 200x^4 - 60x^3 - 69x^2 + 13x - 139 \), it becomes a matter of handling large numbers proficiently. Calculations are done individually for each term:

• Calculate \( 6 \times 7^7 = 6 \times 823543 \), and continue similarly for all terms.
• Add and subtract these results according to the expression format.

This method is reliable because it directly mirrors the mathematical definition of a polynomial. However, it requires careful handling of potentially large intermediate results to ensure accuracy. Direct substitution provides the same final result of \( P(7) = 1 \), confirming the precision of synthetic division as well.