Problem 20 - Precalculus : Building Concepts and Connections

Question

Write each polynomial in the form \(p(x)=d(x) q(x)+r(x),\) where \(p(x)\) is the given polynomial and \(d(x)\) is the given factor. You may use synthetic division wherever applicable. $$-x^{6}+4 x^{5}-x^{3}+x^{2}+x-8 ; x^{2}+4$$

Step-by-Step Solution

Answer

The final result, written in the form \(p(x)=d(x) q(x)+r(x)\) is \(-x^{6}+4x^{5}-x^{3}+x^{2}+x-8 = (x^{2}+4)(-x^{4} + x + 2) -8x -8\). So in this case, \(q(x) = -x^{4} + x + 2\) and \(r(x) = -8x -8\).

Step 1: Set up the long division

Set up the polynomial long division as you would with numbers. Draw a division symbol and write the denominator \(d(x) = x^{2}+4\) to the left of the division symbol. To the right, write the numerator \(p(x) = -x^{6}+4x^{5}-x^{3}+x^{2}+x-8\). Be sure to align like terms vertically, adding zeros where necessary to represent any missing terms, which in this case, there isn't any.

Step 2: Perform the division

Divide the first term of the numerator by the first term of the denominator. This gives \(-x^{4}\) which is the first term of the quotient \(q(x)\). Multiply \(d(x)\) by \(-x^{4}\) and subtract the result from \(p(x)\) to get the intermediate result \(x^{5}+4x^{4}-x^{3}+x^{2}+x-8\). Repeat this process with the intermediate result and the divisor \(d(x)\). Continue this process until you have a remainder whose degree is less than the degree of the divisor.

Step 3: Identify the quotient and remainder

At the end of the division process, the terms that you've collected form the quotient \(q(x)\) and the final result obtained is the remainder \(r(x)\)

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials when the divisor is a linear polynomial, typically of the form (x - c). This technique significantly reduces the amount of writing and calculation needed compared to polynomial long division.

Synthetic division involves setting up a row of coefficients from the dividend polynomial. Below this, a row is created where multiples of the divisor's root are added to these coefficients, starting from the left. Each new value is multiplied by the root and added to the subsequent coefficient in the dividend. The final row of numbers represents the coefficients of the quotient polynomial, with the last number being the remainder.

However, this method isn't applicable to all types of division problems. For instance, in our original exercise, d(x) = x^2 + 4 is not linear; thus, we need to stick with polynomial long division.

Polynomial Division

Polynomial division is similar to long division with numbers, but it handles dividing polynomials rather than integers. It involves dividing a polynomial (the dividend) by another polynomial (the divisor), resulting in a quotient and possibly a remainder.

The procedure includes multiple steps: arranging the dividend and divisor, dividing the leading terms, multiplying the divisor by the result of that division to subtract from the dividend, and repeating these steps until you get a remainder that is either zero or has a degree lower than that of the divisor.

This process can be daunting at first, but understanding that each step simply applies basic algebra rules can make it seem less complex. Remember: every subtraction step in polynomial division is really about zeroing out the highest degree term of the current dividend.

Dividing Polynomials

Dividing polynomials is a foundational skill in algebra that helps solve a variety of mathematical problems, including simplifying rational expressions and solving polynomial equations. To divide polynomials:

• Arrange the terms in descending powers of x,
• If any terms are missing, fill in with zeros,
• Divide the leading term of the dividend by the leading term of the divisor,
• Multiply the entire divisor by this result and subtract from the dividend,
• Repeat the process with the new, reduced polynomial that results.

Polynomial division's ultimate goal is to simplify the given polynomial into a form that is more manageable and understandable. It is crucial to keep track of the degrees of the polynomials to know when the process is complete, which is when you cannot divide the terms any further without increasing the degree of the remainder above that of the divisor.

Remainder Theorem

The Remainder Theorem is a result in algebra that provides a shortcut to finding the remainder of a polynomial when it’s divided by a linear divisor of the form (x - c). The theorem states that if a polynomial f(x) is divided by (x - c), the remainder is f(c).

This means you don't need to fully perform the division if you're only interested in the remainder; simply evaluate the polynomial at c. The Remainder Theorem is also useful when it comes to understanding the results of synthetic division, and it can be effectively applied when the divisor is of the first degree. However, for higher-degree divisors, such as the one in our original problem (d(x) = x^2 + 4), the Remainder Theorem does not directly apply, and we must rely on general polynomial long division to find the remainder.