Problem 8 - Precalculus

Question

Use long division to find the quotients and the remainders. Also, write each answer in the form \(p(x)=d(x) \cdot q(x)+R(x),\) as in equation (2) in the text. $$\frac{4 x^{3}-x^{2}+8 x-1}{x^{2}-x+1}$$

Step-by-Step Solution

Answer

Quotient: \(4x + 3\), Remainder: \(7x - 4\).

Step 1: Setup the Long Division

Arrange the polynomials in long division format. Place the dividend, \(4x^3 - x^2 + 8x - 1\), under the division symbol and the divisor, \(x^2 - x + 1\), outside the division symbol.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(x^2\), which results in \(4x\). This is the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the entire divisor \(x^2 - x + 1\) by the quotient term \(4x\) to get \(4x^3 - 4x^2 + 4x\). Subtract this result from the dividend \(4x^3 - x^2 + 8x - 1\) to get a new polynomial: \(3x^2 + 4x - 1\).

Step 4: Repeat Division with New Polynomial

Divide the leading term of the new polynomial \(3x^2\) by the leading term of the divisor \(x^2\), resulting in \(3\). Add \(3\) to the quotient.

Step 5: Multiply and Subtract Again

Multiply the divisor \(x^2 - x + 1\) by the new quotient term \(3\) to get \(3x^2 - 3x + 3\). Subtract this from \(3x^2 + 4x - 1\) to get the remainder \(7x - 4\).

Step 6: Write the Final Result

The quotient is \(4x + 3\) and the remainder is \(7x - 4\). Therefore, the polynomial expression can be written as: \[4x^3 - x^2 + 8x - 1 = (x^2 - x + 1)(4x + 3) + (7x - 4)\]

Key Concepts

Long Division

Long division in polynomials is a method similar to dividing numbers. It involves dividing the dividend by the divisor to obtain a quotient and a remainder. This technique helps simplify polynomials by breaking them down into easier components.

In our example, we divide the polynomial \(4x^3 - x^2 + 8x - 1\) (the dividend) by \(x^2 - x + 1\) (the divisor).

• Write the dividend underneath the long division bar.
• Place the divisor to the outside left of this setup.

This setup prepares us to systematically subtract multiples of the divisor using terms from the dividend, simplifying the division process. Remember, you're always working with matching degrees to methodically reduce the polynomial.

Remainder Theorem

The remainder theorem is a useful concept in polynomial division. It states that for any polynomial \(p(x)\) divided by \(x - c\), the remainder of this division is the value \(p(c)\).

While this theorem is typically used for dividing by linear factors, understanding it helps us grasp what happens with remainders. After dividing a polynomial using long division, we are left with a simpler polynomial (the quotient) and whatever is left over is termed the remainder.

• In our example, after performing the long division, we have a remainder of \(7x - 4\).
• This remainder represents what couldn't be divided evenly by our divisor \(x^2 - x + 1\).

The remainder theorem reassures us that if the remainder was substituted back into our original polynomial process at a point \(x = c\), it would equate back to this remainder value.

Dividend and Divisor

In the process of polynomial division, the term "dividend" refers to the polynomial that you're dividing. The "divisor" is what you are dividing by. These are essential components in setting up and carrying out the division process.

In the example problem:

• The dividend is \(4x^3 - x^2 + 8x - 1\).
• The divisor is \(x^2 - x + 1\).

Understanding which polynomial is which helps in arranging the division properly. The degree of the dividend should be larger or equal to that of the divisor, making it possible to perform the long division correctly. Dividing polynomials is simply like a more complex version of dividing numbers.

Quotient and Remainder

When dividing polynomials, the result consists of a "quotient" and a "remainder". The quotient is the result of the division, while the remainder is what is left over. The division can be expressed using these terms as \(p(x) = d(x) \cdot q(x) + R(x)\).

In the context of our exercise, after performing the redistribution of terms:

• The quotient is \(4x + 3\).
• The remainder is \(7x - 4\).

This expression follows the rule
\[ 4x^3 - x^2 + 8x - 1 = (x^2 - x + 1)(4x + 3) + (7x - 4) \]
By verifying each term through multiplication and addition, one can ensure the accuracy of this division. This result shows how polynomials can be simplified and analyzed through division.