Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use long division to divide. $$\left(8 x^{4}-5\right) \div(2 x+1)$$

Step-by-Step Solution

Answer

The quotient obtained from the long division of the given polynomials is \(4x^{3} - 2x^{2} + x + 1\).

Step 1: Write Division as Fraction

First, write the division as a fraction. It will look like this: \(\frac{8 x^{4}-5}{2 x+1}\).

Step 2: Set Up Division

Next, set the division up as you would in the traditional format. Write \(2x + 1\) on the outside of the box and write \(8x^{4} - 5\) on the inside of the box.

Step 3: Divide the first term of the numerator by the first term of the denominator

Division of \(8x^{4}\) by \(2x\), will give \(4x^{3}\). Go ahead and write that above the line.

Step 4: Multiply and Subtract

Now multiply \(4x^{3}\) times \(2x + 1\), then subtract this value from the numerator \(8x^{4} - 5\). This should yield \(0x^{4} - 4x^{3} - 5\). Bring down the next term if any.

Step 5: Repeat Dividing, Multiplying and Subtracting

Divide the new first term in the numerator by the first term of the denominator, then multiply and subtract as before. After these steps, the obtained numerator will be \(0x^{2} - 2x - 5\).

Step 6: Repeat Steps Again

As the degree of in the new numerator is higher than in the denominator, continue the division process. This will create the term \(x^{2}\) in our answer.

Step 7: Final cycle

By going one more cycle through these steps, the numerator will become \(x + 1\). The first term's degree in the numerator is now equal to that in the denominator.

Step 8: The left numerator after last subtraction

After the last subtraction, we are left with \(1 - 1 = 0\) . As the degree of this last expression is less than in the denominator, the process stops here.

Step 9: Writing the Final Answer

The final answer will be \(4x^{3} - 2x^{2} + x + 1\).

Key Concepts

Long Division Method

The long division method is a technique used to divide polynomials, similar to how numbers are divided in arithmetic. It involves a step-by-step process to ensure precise calculation. This method is essential when dividing one polynomial by another, especially when they don't factor easily or have complex coefficients. To apply the long division method effectively, follow these general steps:

• Write the polynomial you're dividing (the dividend) and the polynomial you're dividing by (the divisor) in a long division setup, with the divisor outside and the dividend inside the division bar.
• Determine how many times the leading term of the divisor can be multiplied to match the leading term of the dividend. Write this result above the division bar.
• Multiply the entire divisor by this result and subtract it from the dividend.
• Repeat the process with the newly formed polynomial obtained after subtraction. Continue until the degree of the remainder is less than the degree of the divisor.

This method gives us both the quotient and possibly a remainder. It's a reliable technique, even when the polynomials involved are complex.

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of variables and coefficients, involving only the operations of addition, subtraction, and multiplication. They follow the structure \[ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \], where:

• \(a_n, a_{n-1}, \ldots, a_0\) are coefficients.
• \(n\) indicates the power or degree of each term.
• \(x\) is the variable.

In polynomial division, understanding polynomial expressions helps identify terms that need to be addressed during division. Each polynomial's terms are sorted by their degree in descending order, allowing for systematic processing during division. Polynomials are versatile and appear in various forms, such as binomials with two terms or trinomials with three terms, and they are used in various math and science applications.

Degree of Polynomials

The degree of a polynomial is the highest power of its variable with a non-zero coefficient. Understanding the degree is crucial in polynomial division because it controls the division process. When applying the long division method:

• The degree of the dividend should be compared to the degree of the divisor to see if division can continue. Division stops when the degree of the remainder is less than the degree of the divisor.
• Tracking changes in the degree throughout the division process helps to understand how many cycles are needed to complete the calculation.

For example, with the polynomial \(8x^4 - 5\), the degree is 4. When dividing by \(2x + 1\) with a degree of 1, the long division is set up in multiple steps to reduce the degree of the numerator systematically. Understanding the degree ensures that each subtraction step correctly reduces the polynomial until a remainder smaller than the divisor's degree is achieved. This understanding underlines why certain steps are repeated more than once and helps in predicting the form of the quotient efficiently.