Problem 50 - Algebra Form and Function

Question

Consider the polynomial \(x^{5}-3 x^{4}+4 x^{3}-2 x+1\) (a) What is the value of the polynomial when \(x=4\) ? (b) If \(a\) is the answer you found in part (a), show that \(x-4\) is a factor of \(x^{5}-3 x^{4}+4 x^{3}-2 x+1-a\)

Step-by-Step Solution

Answer

Question: Evaluate the polynomial \(x^{5}-3 x^{4}+4 x^{3}-2 x+1\) when x=4 and prove that x-4 is a factor of the polynomial minus a. Answer: When x=4, the polynomial evaluates to 505. By using synthetic division, we have shown that x-4 is a factor of the polynomial minus a, which is given by \((x-4)(x^{4}-2x^{3}+6x^{2}-14x+50)=x^{5}-3 x^{4}+4 x^{3}-2 x+1-505\).

Step 1: Evaluate the polynomial when x=4

Plug in x=4 into the polynomial, and find the result: \(P(4)=4^{5}-3(4^{4})+4(4^{3})-2(4)+1\) Calculate the result: \(P(4)=1024-3(256)+4(64)-8+1=1024-768+256-8+1=505\)

Step 2: Substitute a with the result found in step 1

Now we have to replace a with the result we found in step 1: \(x^{5}-3 x^{4}+4 x^{3}-2 x+1-a= x^{5}-3 x^{4}+4 x^{3}-2 x+1-505\)

Step 3: Check if x-4 is a factor of the polynomial minus a using synthetic division

Use synthetic division to check if x-4 is a factor of the polynomial: \((x-4)(x^{4}+bx^{3}+cx^{2}+dx+e)\) Divide the polynomial by \((x-4)\): ``` ________________ 4 | 1 -3 4 -2 0 1-505 1 1 -2 -12 50 ------------------------------------- 0 -2 6 -14 50 -504 ``` As the remainder is 0, \((x-4)\) is a factor of the polynomial minus a. Therefore, we can conclude that: \((x-4)(x^{4}-2x^{3}+6x^{2}-14x+50)=x^{5}-3 x^{4}+4 x^{3}-2 x+1-505\)

Key Concepts

Factorization

Factorization involves breaking down a polynomial into simpler components or factors that, when multiplied together, give back the original polynomial. It's like dividing a number into its prime factors such as 12 into 2, 2, and 3. For example, finding factors can help simplify complex equations.
When we determine if a linear expression like \(x-4\) is a factor, it implies that dividing our polynomial by \(x-4\) should leave no remainder. This is confirmed using polynomial long division or synthetic division. When there is no remainder, \(x-4\) is indeed a factor.

• Factors make solving and understanding polynomials simpler.
• The goal is to express the polynomial as a product of simpler polynomials.
• Effective factorization helps in polynomial evaluation and solving equations.

In cases where \(x = 4\), substituting back into the polynomial confirms if the remainder is zero, proving that \(x-4\) is a factor.

Synthetic Division

Synthetic division is a shortcut method for dividing polynomials, particularly useful for dividing by expressions of the form \(x - c\). It's more efficient compared to the traditional polynomial division.
Here's how synthetic division works:

• Write down the coefficients of the polynomial.
• Use the value of \(c\) from \(x-c\) (e.g., \(x-4\) means \(c=4\)).
• Perform the division using simple arithmetic operations, bringing down and multiplying. Add the results as you move across.
• If the remainder (the final number) is zero, then \(x-c\) is a factor.

This technique was applied to confirm the factor \(x-4\) in the exercise. The table in the original solution shows each step of synthetic division, providing a clear trail of how the final remainder proved that \(x-4\) was indeed a factor of our adjusted polynomial.

Polynomial Evaluation

Polynomial evaluation is calculating the value of a polynomial when a specific value is substituted for the variable, often denoted as \(P(x)\). In the exercise, evaluating the polynomial at \(x=4\) involved substituting 4 into the polynomial and simplifying to find \(P(4)\). This helps determine specific points on the polynomial curve.
Here's how you evaluate a polynomial:

• Replace the variable with the given number (4 in this case).
• Use order of operations: calculate powers, then multiply and add/subtract.
• Carry out calculations step-by-step to avoid mistakes.

By determining \(P(4) = 505\), this value was used to adjust the polynomial, effectively helping factor the polynomial as required. Polynomial evaluation is a fundamental tool for analyzing the behavior and graph of polynomials.