Problem 308 - Precalculus

Question

Use long division to divide. Specify the quotient and the remainder. $$\left(6 x^{2}-25 x-25\right) \div(6 x+5)$$

Step-by-Step Solution

Answer

Quotient: \(x - 5\), Remainder: 0.

Step 1: Set Up the Long Division

Write the dividend \(6x^2 - 25x - 25\) and the divisor \(6x + 5\) in long division format. Align the terms of the polynomial according to their degree in descending order.

Step 2: Divide the First Terms

Divide the first term of the dividend \(6x^2\) by the first term of the divisor \(6x\), which gives \(x\). This is the first term in the quotient.

Step 3: Multiply and Subtract

Multiply the entire divisor \(6x + 5\) by \(x\), yielding \(6x^2 + 5x\). Subtract this from the original dividend \(6x^2 - 25x - 25\), resulting in a new polynomial \(-30x - 25\).

Step 4: Repeat Division for Next Term

Divide the new leading term \(-30x\) by \(6x\), giving \(-5\). This is the next term in the quotient.

Step 5: Multiply and Subtract Again

Multiply the divisor \(6x + 5\) by \(-5\), resulting in \(-30x - 25\). Subtract this from \(-30x - 25\) to get a remainder of 0.

Step 6: Write Final Answer

The quotient of the division is \(x - 5\) and the remainder is 0. Therefore, \(6x^2 - 25x - 25 = (6x + 5)(x - 5)\).

Key Concepts

Polynomial Division

When dividing polynomials, one efficient method is called long division. This is similar to numerical long division. For a polynomial division, you have a dividend (the polynomial you're dividing) and a divisor (the polynomial you are dividing by).

To conduct polynomial division:

• Arrange both the dividend and divisor in standard form, which means ordering from highest degree to lowest.
• Divide the term with the highest degree in the dividend by the term with the highest degree in the divisor.
• The result is the first term of the quotient. Multiply the entire divisor by this term and subtract the result from the dividend.
• Repeat with the new polynomial until the degree of the new polynomial is less than the degree of the divisor.

By following these steps, you will be able to find the quotient and the remainder for your division.

Quotient and Remainder

In polynomial division, the quotient and remainder play crucial roles in understanding the relationship between the original polynomials. The quotient is what you get when you divide the dividend entirely by the divisor.

In our specific example:

• The quotient tells us that if we multiply it by the divisor, adding any remainder, will return us back to the original dividend.
• In the case of \(\left(6 x^2 - 25 x - 25\right) \div (6 x + 5)\), we found that the quotient is \(x - 5\).

The remainder, which in this example is 0, indicates the dividend is exactly divisible by the divisor without any leftover amount. The division can be expressed as an equation:\[6x^2 - 25x - 25 = (6x + 5)(x - 5) + 0\]This equation reflects the precise relationship between the parts, with the remainder confirming that there is no left-over value after the division.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication). Understanding these expressions is key to manipulating them correctly during polynomial division.

When dealing with algebraic expressions in the context of polynomial division, keep in mind:

• Each term in the expression is made up of constants (numbers) and variables with certain powers.
• Align the polynomials by their degree during division, as this ensures each part is successfully divided without error.
• The operations used in division such as subtraction and multiplication must respect algebraic rules.
• The goal is to cancel out terms gradually, stepping down the degrees until the remainder is either zero or of a lower degree than the divisor.

Grasping these features of algebraic expressions enhances your ability to solve polynomial division problems effectively.