Problem 52 - 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

\(P(7) = 1\) using both synthetic division and substitution.

Step 1: Understanding Synthetic Division

To evaluate \(P(7)\) using synthetic division, we use \(x = 7\) as our divisor, essentially treating it as \(x-7\). We set up a row for the coefficients of \(P(x)\) which are: [6, -40, 16, -200, -60, -69, 13, -139].

Step 2: Performing Synthetic Division

Write down the coefficients: 6, -40, 16, -200, -60, -69, 13, -139. We start by bringing down the leading coefficient (6) as is. Multiply by the divisor (7) and add to the next coefficient: 1. Bring down the 6.2. Multiply 6 by 7 = 42, add to -40 to get 2.3. Multiply 2 by 7 = 14, add to 16 to get 30.4. Multiply 30 by 7 = 210, add to -200 to get 10.5. Multiply 10 by 7 = 70, add to -60 to get 10.6. Multiply 10 by 7 = 70, add to -69 to get 1.7. Multiply 1 by 7 = 7, add to 13 to get 20.8. Multiply 20 by 7 = 140, add to -139 to get 1.The last number, 1, is \(P(7)\).

Step 3: Substituting and Evaluating Directly

Substitute \(x = 7\) directly into \(P(x)\):\[ P(7) = 6(7)^7 - 40(7)^6 + 16(7)^5 - 200(7)^4 - 60(7)^3 - 69(7)^2 + 13(7) - 139 \]Calculate step by step:- \(6 \times 7^7 = 6 \times 823543 = 4941258\)- \(-40 \times 7^6 = -40 \times 117649 = -4705960\)- \(16 \times 7^5 = 16 \times 16807 = 268912\)- \(-200 \times 7^4 = -200 \times 2401 = -480200\)- \(-60 \times 7^3 = -60 \times 343 = -20580\)- \(-69 \times 7^2 = -69 \times 49 = -3381\)- \(13 \times 7 = 91\)- Remaining \(-139\)Sum these values: \(4941258 - 4705960 + 268912 - 480200 - 20580 - 3381 + 91 - 139 = 1\). Thus, \(P(7) = 1\).

Step 4: Conclusion

Both methods, synthetic division and direct substitution, yield \(P(7) = 1\).

Key Concepts

Polynomial Evaluation

When talking about polynomial evaluation, we're considering the process of finding the value of a polynomial function at a specific point. Think of a polynomial like \( P(x) \), full of various terms. To evaluate it, we plug in a number for \( x \). This means replacing every instance of \( x \) with this specific number and calculating the result.

Polynomial evaluation is handy when we need to determine the polynomial's output at particular values without graphing or deeper analysis. In the given exercise, the task is to evaluate \( P(x) \) at \( x = 7 \). We aim to find \( P(7) \), which is precisely the value the polynomial takes when \( x \,= 7 \).

• This involves either substituting 7 into the polynomial directly or using a technique like synthetic division to simplify the process.

Both methods should lead us to the same result, validating our solution.

Direct Substitution

Direct substitution is a straightforward method to evaluate a polynomial function. It's exactly what it sounds like: substituting a specific value in place of \( x \) in a polynomial expression.

Consider a polynomial \( P(x) \). For example, if we want to find \( P(7) \), we replace every \( x \) in the expression with 7, then calculate the resulting values across each term of the polynomial.

In the problem provided, direct substitution for \( P(x) \) yields:

• Replace \( x \) with 7 in each term: \( 6(7)^7 - 40(7)^6 + 16(7)^5 \ldots - 139 \).
• Calculate each power of \( 7 \) and multiply by the corresponding coefficients.
• Sum the results to find the value of the polynomial when \( x = 7 \).

This method can sometimes be cumbersome, especially for high-degree polynomials due to large calculations, but it is highly intuitive.

Polynomial Coefficients

Polynomial coefficients are the numerical factors that multiply each term of a polynomial. When we look at a polynomial like \( P(x) \), each term's coefficient plays a crucial role in defining the polynomial's behavior and its resulting graph.

In the standard form, a polynomial can be expressed as:

• \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0 \)

where \( a_n, a_{n-1}, \ldots, a_0 \) are the coefficients.

For the polynomial given in the exercise, the coefficients are \([6, -40, 16, -200, -60, -69, 13, -139]\). These determine how strongly each power of \( x \) influences the output of \( P(x) \).

• Higher coefficients have a more significant impact on the polynomial's behavior as \( x \) increases or decreases.
• Understanding coefficients helps in performing operations like synthetic division effectively.

In synthetic division, identifying and using these coefficients efficiently allows for a streamlined evaluation process.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of a variable multiplied by coefficients. They form the backbone of algebra and calculus due to their polynomial nature, making them foundational in many complex equations and systems.

The general form of a polynomial is expressed as \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0 \), where the powers of \( x \) decrease and each term is scaled by a coefficient.

Polynomial functions are characterized by:

• Having whole number powers of \( x \).
• Each term consisting of a coefficient and a variable part.
• The degree of the polynomial, which is determined by the highest power of \( x \).

In the exercise, \( P(x) \) is a seventh-degree polynomial (since the highest power is 7).

Polynomials are continuous, smooth, and can be analyzed for critical points, roots, and intervals of increase/decrease. They are widely used for modeling real-world situations and understanding changes across different parameters.