Question

Given \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{8}+2 \mathrm{x}^{7}-3 \mathrm{x}^{6}+\mathrm{x}^{5}-2 \mathrm{x}^{4}-4 \mathrm{x}^{3}+\mathrm{x}^{2}-\mathrm{x}+6\) Compute the remainder, in this case \(\mathrm{f}(3)\). (Hint: use the remainder theorem and synthetic division,)

Short answer

The remainder when the function \(f(x) = x^8+2x^7-3x^6+x^5-2x^4-4x^3+x^2-x+6\) is divided by \( x - 3 \) is 8844. This is calculated using synthetic division and the Remainder theorem.

Step-by-step solution

Understand the task

The task is to find the remainder of the given function when divided by \( (x - 3) \), which is simply equivalent to finding \( f(3) \) due to the Remainder theorem.

Write down the function and identify coefficients

We need to find the coefficients of the function. Here, the function \( f(x) \) in descending order of degree is given by \[ f(x) = x^8+2x^7-3x^6+x^5-2x^4-4x^3+x^2-x+6 \] The coefficients are 1, 2, -3, 1, -2, -4, 1, -1, 6.

Start synthetic division

Draw a horizontal line, write the coefficients above the line. On the far left below the line, write the value for which we're finding \( f \), in this case 3. Then start synthetic division: carry the first coefficient straight down and multiply it by 3, writing the result under the second coefficient. Add these values together and carry the result down. Repeat this process until you reach the end.

Do synthetic division

Following the process outlined in step 3: \[ \begin{array}{c|ccccccccc} 3 & 1 & 2 & -3 & 1 & -2 & -4 & 1 & -1 & 6 \\ & & 3 & 15 & 36 & 111 & 327 & 981 & 2946 & 8838 \\ \hline & 1 & 5 & 12 & 37 & 109 & 323 & 982 & 2945 & 8844 \end{array} \]

Interpret the result

The numbers on the bottom row (excluding the far right value) are the coefficients of the division result. The far right number is the remainder. In this case, the remainder is 8844. Therefore, when the function \( f(x) \) is divided by \( x - 3 \), the remainder is 8844. This means that if we replace x with 3 in the original polynomial, it gives 8844, so \( f(3) = 8844 \).

Key concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form \(x - c\), where \(c\) is a constant. Unlike long division, synthetic division is more efficient and involves fewer operations. It is particularly helpful when you need to find the remainder quickly.
Here is how synthetic division works:

• Start by writing down the coefficients of the polynomial in sequence, even if some coefficients are zero.
• Write the value of \(c\) from \(x - c\) on the left, outside of the main division area.
• The leading coefficient is carried down as it is.
• Multiply it by \(c\), the value on the left, and write the result below the next coefficient.
• Add the numbers and repeat the multiply-add process until all coefficients have been involved.

The last number in the bottom row of synthetic division is the remainder of the division, which also corresponds to the value of the polynomial evaluated at \(c\), thanks to the Remainder Theorem.

Polynomial Division

Polynomial division is a process used to divide one polynomial by another. Traditional polynomial division, often called long division when applied to polynomials, is similar to dividing numbers. It involves:

• Aligning terms by degree.
• Subtracting products to remove leading terms one step at a time.
• Bringing down subsequent terms after each division step to form a new dividend.

Although more tedious than synthetic division, polynomial division is more applicable to divisors that are not simply linear. It is essential for obtaining quotient polynomials or understanding the structure of polynomial expressions.

Evaluating Polynomials

Evaluating a polynomial at a particular value involves calculating the output of the polynomial function for a given input. For instance, if you have a polynomial \(f(x)\), evaluating it at \(x = c\) means calculating \(f(c)\), which is essentially replacing every instance of \(x\) in the polynomial with \(c\) and simplifying.

The Remainder Theorem provides a handy way to evaluate polynomials. It states that the remainder of a polynomial \(f(x)\) divided by \(x - c\) is \(f(c)\). This principle allows synthetic division not just to determine the quotient, but also to directly calculate the remainder, which is \(f(c)\). Using synthetic division for this purpose is efficient, especially useful for higher-degree polynomials.

Polynomial Coefficients

Polynomial coefficients are the numbers that multiply the terms of a polynomial. For example, in the polynomial \(f(x) = x^8 + 2x^7 - 3x^6 + x^5 - 2x^4 - 4x^3 + x^2 - x + 6\), the coefficients are \(1, 2, -3, 1, -2, -4, 1, -1, 6\). Each coefficient corresponds to a specific power of \(x\), starting from the highest power and going down to the constant term.

Coefficients play a crucial role in operations involving polynomials. For instance, in synthetic division, we rely on these coefficients to perform arithmetic operations rapidly without dealing with powers of \(x\). Identifying and correctly ordering polynomial coefficients are essential steps in both synthetic division and polynomial evaluation.