Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the quotient and remainder: \(x^{3}-7 x^{2}+19 x-15\) is divided by \(x-3\)

Short answer

Quotient: \(x^2 - 4x + 7\), Remainder: \(6\)

Step-by-step solution

Set up the Division

Write the polynomial as the dividend and the linear binomial as the divisor. Here, the dividend is \(x^3 - 7x^2 + 19x - 15\) and the divisor is \(x - 3\).

Perform Polynomial Long Division

Divide the first term of the dividend \(x^3\) by the first term of the divisor \(x\), which gives \(x^2\). Write \(x^2\) above the division line.

Multiply and Subtract

Multiply \(x^2\) by \(x - 3\), which results in \(x^3 - 3x^2\). Subtract this from the dividend to get a new polynomial: \(-4x^2 + 19x - 15\).

Repeat the Process

Divide the new leading term \(-4x^2\) by \(x\), which gives \(-4x\). Write \(-4x\) above the division line next to \(x^2\).

Multiply and Subtract Again

Multiply \(-4x\) by \(x - 3\), resulting in \(-4x^2 + 12x\). Subtract this from the current polynomial to get a new polynomial: \(7x - 15\).

Final Division Step

Divide \(7x\) by \(x\), which results in \(7\). Write \(7\) above the division line next to \(-4x\).

Final Multiplication and Subtraction

Multiply \(7\) by \(x - 3\), giving \(7x - 21\). Subtract this from the current polynomial to get the remainder: \(6\).

Write the Final Answer

The quotient from the polynomial division is \(x^2 - 4x + 7\), and the remainder is \(6\).

Key concepts

Polynomial Division

Polynomial division is like long division for numbers, but it works with algebraic expressions. Just like in arithmetic, we divide, multiply, subtract, and repeat the process. This method helps break down complex polynomials into simpler parts. For our example, we want to divide the polynomial \(x^{3} - 7x^{2} + 19x - 15\) by \(x - 3\). It's a systematic process that reveals both a quotient and a remainder. This way, we can handle even those large, complicated expressions one step at a time.

Quotient and Remainder

When dividing polynomials, we get two main results: the quotient and the remainder. The quotient is what we get from the division process, while the remainder is what's left after all possible divisions are done. For our equation, we found that when \(x^{3} - 7x^{2} + 19x - 15\) is divided by \(x - 3\), the quotient is \(x^{2} - 4x + 7\) and the remainder is \6\. This is written mathematically as:
\[ x^{3} - 7x^{2} + 19x - 15 = (x - 3)(x^{2} - 4x + 7) + 6 \] This shows how the polynomial was broken down, leaving us with a simpler expression plus a small leftover part.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operations like addition, subtraction, multiplication, and division. They can represent real-world problems and can be simplified through various methods. In polynomial division, we work with expressions like \(x^{3} - 7x^{2} + 19x - 15\). Understanding how to manipulate these expressions is key to solving a wide range of algebraic problems. Algebraic expressions are the backbone of algebra, and mastering them allows us to progress to more advanced math topics.

Step-by-Step Solution

Breaking down a problem into manageable steps is essential in mathematics. Let's look at our polynomial division in detail.

• Step 1: Set up the polynomial and divisor.
• Step 2: Divide the first terms: \(x^{3} \div x = x^2\).
• Step 3: Multiply and subtract: \(x^{2}(x - 3) = x^{3} - 3x^{2}\), subtract to get \ -4x^{2} + 19x - 15\.
• Step 4: Repeat: \ -4x^{2} \div x = -4x\, then multiply and subtract: \ -4x(x - 3) = -4x^{2} + 12x\, subtract to get \ 7x - 15\.
• Step 5: Final steps: \ 7x \div x = 7\, then multiply and subtract: \ 7(x - 3) = 7x - 21\, subtract to get remainder \ 6\.

By following each step carefully, we ensure that our solution is accurate and clear.