Problem 43 - Algebra A Combined Function

Question

Mixed Practice Divide. If the divisor contains 2 or more terms, use long division. See Examples 1 through 8. $$ \frac{5-6 x^{2}}{x-2} $$

Step-by-Step Solution

Answer

Quotient is \(-6x - 12\), remainder is \(-19\).

Step 1: Set Up the Division

Since the divisor has only one term, we don't need to rearrange anything. We have: Dividend = \(5 - 6x^2\) and divisor = \(x-2\).

Step 2: Express in Long Division Form

Write the dividend and divisor in long division format: \( (x-2)\) outside the division bracket and \(5-0x-6x^2\) inside.

Step 3: Divide the First Terms

Divide the highest degree term of the dividend by the highest degree term of the divisor: \(-6x^2 \div x = -6x\). Place \(-6x\) above the division bracket.

Step 4: Multiply and Subtract

Multiply \(-6x\) by \(x-2\) to get \(-6x^2 + 12x\). Write this below the dividend and subtract from the dividend to get \(-12x + 5\).

Step 5: Repeat Division

Divide the term \(-12x\) by \(x\) to get \(-12\). Place \(-12\) next to \(-6x\) above the division bracket.

Step 6: Multiply and Subtract Again

Multiply \(-12\) by \(x-2\) to get \(-12x + 24\). Subtract this from \(-12x + 5\) to get \(-19\).

Step 7: Write the Remainder

Since the degree of the remainder \(-19\) is less than the degree of the divisor, we can stop here. The final result is \(-6x - 12\) with a remainder of \(-19\).

Key Concepts

Understanding Long Division in Polynomials

When it comes to dividing polynomials, long division is a reliable method especially when the divisor has two or more terms. Think of it as a process similar to long division of numbers, but with algebraic expressions instead. The goal is to divide the polynomial (dividend) by another polynomial (divisor) to find the quotient and possibly a remainder.

• First, you arrange the polynomials in descending order of their degrees.
• Next, you identify the leading terms of both the dividend and the divisor.
• Then, you proceed with the division by focusing on the highest degree terms first.

The process involves multiple steps of dividing, multiplying, subtracting, and bringing down lower degree terms, much like traditional arithmetic division, until the remainder is of a lower degree than the divisor.

The Role of the Dividend in Polynomial Division

In polynomial division, the dividend is the polynomial you are dividing. It is typically written inside the division bracket during the long division setup. For our example, the dividend is expressed as the polynomial: \[ 5 - 6x^2 = 0x^1 + 5x^0 - 6x^2 \].

• Notice that all terms are arranged in a descending order according to their degree.
• If any degrees are missing in the sequence, they are filled with a zero coefficient. This makes subtraction easier during the division process.

Understanding the structure of the dividend is essential. It ensures that you can correctly align terms when performing the division, which reduces any chance of errors and maintains accuracy in calculations.

Divisor: Key to Dividing

The divisor is the polynomial that you divide the dividend by. It is typically placed outside the division bracket in the long division format. In our scenario, the divisor is simple: \[ x - 2 \].

• The degree of the divisor is the key in determining when to stop the division. You stop once the degree of the remainder is less than the degree of the divisor.
• Focus on the leading term of the divisor while performing the division step, as it helps determine what to multiply with the quotient.

The ease of division largely depends on the structure of the divisor. Managing it carefully, especially in terms of its leading coefficient and degree, will streamline the entire process.

What Remainder Tells Us

The remainder is what is left after performing polynomial division, and it plays a vital role in expressing the entire division result. When the division process is complete, the remainder is the polynomial that has a degree less than that of the divisor. In the case we worked through, the remainder is:\[ -19 \].

• The remainder shows the difference that could not fit perfectly into the divisor.
• It is important in understanding the accuracy and completeness of the division.
• In many cases, the division result can be expressed as: quotient + (remainder/divisor).

If the remainder is zero, the divisor is exactly a factor of the dividend. Otherwise, it represents the leftover difference.