Problem 2 - Beginning and Intermediate Algebra

Question

Label the dividend, divisor, and quotient of each division problem. $$2 p + 3 \longdiv { 1 0 p ^ { 3 } + p ^ { 2 } - 2 5 p - 6 } ^ { 5 p ^ { 2 } - 7 p - 2 }$$

Step-by-Step Solution

Answer

Short Answer: Dividend: \(10p^3 + p^2 - 25p - 6\) Divisor: \(2p + 3\) Quotient: \(5p^2 - 7p - 3\)

Step 1: Set up the long division

Set up the long division problem as follows: \( \begin{array}{c|cccc} \multicolumn{2}{r}{5p^2} & -7p & -3 \\ \cline{2-5} 2p + 3 & 10p^3 & + p^2 & -25p & -6 \\ \cline{2-5} \end{array} \) Step 2: Divide

Step 2: Perform the division

Multiply the divisor (2p + 3) by the quotient (5p^2 - 7p - 3) and subtract it from the dividend (10p^3 + p^2 - 25p - 6): \( \begin{array}{c|cccc} \multicolumn{2}{r}{5p^2} & -7p & -3 \\ \cline{2-5} 2p + 3 & 10p^3 & + p^2& -25p & -6 \\ & - & (10p^3) & +(3p^2) & -15p \\ \cline{2-5} & & & (-2p^2) & -10p & -6 \\ & & & + & (2p^2) & +3p \\ \cline{4-6} & & & & & -13p & -6 \\ \end{array} \)

Step 3: Identify the components

At this point, we have completed the polynomial long division. We can now identify the components: Dividend: \(10p^3 + p^2 - 25p - 6\) Divisor: \(2p + 3\) Quotient: \(5p^2 - 7p - 3\)

Key Concepts

Dividend

The dividend is a key component in polynomial long division. In our problem, the polynomial we are dividing is the dividend. This is the expression that the remainder and quotient are derived from. Think of the dividend as the total amount you want to divide up, similar to the total number of candies you might distribute among friends. Here, the dividend is \(10p^3 + p^2 - 25p - 6\). This polynomial is what we start with. Our goal is to divide it fully to find out how many times the divisor fits into it, determining the quotient. Understanding the dividend includes knowing what each term represents:

• The term \(10p^3\) is the leading term and primarily influences the division process.
• The coefficients (10, 1, -25, and -6) play a role in the calculations to find both the quotient and any remainder after division.

By breaking down the dividend, you can predict how complex the division might be, as higher-degree terms mean more complex polynomial long division.

Divisor

The divisor is the polynomial by which we divide the dividend in polynomial long division. For this problem, our divisor is \(2p + 3\). Think of the divisor as the number of groups you are making with individuals in division similar to basic arithmetic.The divisor has the following key elements:

• Its degree (highest power of \(p\)) often determines the starting degree of the quotient.
• The leading coefficient (in this case, 2 from \(2p\)) determines how the leading term in the dividend is divided.

During division, we repeatedly measure how many times the divisor can multiply to "consume" sections (terms) of the dividend. The divisor should be arranged in decreasing powers of the variable like any polynomial. This ordering allows for systematic calculation and ensures each step leads gradually to the discovery of the quotient.

Quotient

In polynomial long division, the quotient is what you get when you divide the dividend by the divisor. Think of it as the result or "answer" to your division problem. In our specific exercise, the quotient is \(5p^2 - 7p - 3\). Understanding the quotient involves noticing these traits:

• The quotient represents how many times the divisor fits into the dividend in terms of polynomial degree.
• Each term of the quotient reflects the degree to which parts of the dividend have been divided out.
• The degree of the quotient is determined initially based on the highest power in the dividend divided by the highest power in the divisor.

The quotient informs us of the full portion of the dividend that can be evenly divided by the divisor. The remainder, if any, is then managed separately once the quotient has been determined. Fully comprehending the quotient helps make sense of how the division was achieved and provides a foundation for solving more complex polynomial division problems.