Problem 11 - College Algebra with Modeling and Visualization

Question

Divide the first polynomial by the second. State the quotient and remainder. $$2 x^{4}-7 x^{3}-5 x^{2}-19 x+17 \quad\quad\quad x+1$$

Step-by-Step Solution

Answer

Quotient: \(2x^3 - 9x^2 + 4x - 23\), Remainder: \(40\).

Step 1: Set Up the Division Problem

Write the division problem in the format of polynomial long division. The dividend is \(2x^4 - 7x^3 - 5x^2 - 19x + 17\) and the divisor is \(x + 1\). Placing them under the long division symbol, we set up: \[ \frac{2x^4 - 7x^3 - 5x^2 - 19x + 17}{x + 1} \]

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(2x^4\) by the leading term of the divisor \(x\). This yields \(2x^3\). Write \(2x^3\) above the division line.

Step 3: Multiply and Subtract

Multiply the entire divisor \(x + 1\) by \(2x^3\) to get \(2x^4 + 2x^3\). Subtract this from the original polynomial. The new polynomial becomes: \[-7x^3 - 5x^2 - 19x + 17 - (2x^4 + 2x^3) = -9x^3 - 5x^2 - 19x + 17\]

Step 4: Repeat Division

Repeat the process with the new polynomial. Divide \(-9x^3\) by \(x\) to get \(-9x^2\). Multiply the divisor by \(-9x^2\) resulting in \(-9x^3 - 9x^2\) and subtract it from \(-9x^3 - 5x^2 - 19x + 17\). This yields: \[-5x^2 - 19x + 17 - (-9x^3 - 9x^2) = 4x^2 - 19x + 17\]

Step 5: Continue Division

Divide \(4x^2\) by \(x\), which gives \(4x\). Multiply the divisor by \(4x\) to get \(4x^2 + 4x\), and subtract it from \(4x^2 - 19x + 17\). The new polynomial is: \[-19x + 17 - (4x^2 + 4x) = -23x + 17\]

Step 6: Final Division

Divide \(-23x\) by \(x\), which results in \(-23\). Multiply the divisor \(x + 1\) by \(-23\) to obtain \(-23x - 23\). Subtract this product from \(-23x + 17\) to get the remainder: \[-23x + 17 - (-23x - 23) = 40\]

Step 7: State Quotient and Remainder

The quotient obtained from the division is \(2x^3 - 9x^2 + 4x - 23\) and the remainder is \(40\). The complete solution is: Quotient: \(2x^3 - 9x^2 + 4x - 23\) Remainder: \(40\)

Key Concepts

Long Division

Long division is a method often used to divide polynomials, similar to how you might divide numbers by hand. In polynomial long division, we divide the terms with the highest degrees first, leading to successive simplifications. This involves dividing, multiplying, and subtracting to slowly work through the polynomial.
As we start, the polynomial to be divided is called the dividend, while the polynomial you divide by is the divisor.

• Write both the dividend and divisor under a long division symbol.
• Start with the term in the dividend that has the highest power.
• The result is written as part of the quotient above the division line.
• Multiply the whole divisor by this term of the quotient.
• Subtract the product from the dividend to form a new polynomial.

Repeat these steps using the new polynomial as the dividend until you can no longer divide.

Quotient and Remainder

In polynomial division, much like number division, you get a quotient and sometimes a remainder. The quotient is the result of the division not considering any remainder.
The remainder is a polynomial smaller in degree than the divisor that cannot be divided further by the divisor.

• The quotient for our specific problem is the polynomial obtained during the division: \(2x^3 - 9x^2 + 4x - 23\).
• The remainder is what remains after the division process is complete; here, it's \(40\).

When multiplying the divisor by the quotient and adding the remainder, you should get the original dividend which verifies the division is correct.

Dividing Polynomials

Dividing polynomials can seem complex, but by applying systematic steps, you can solve these problems efficiently. The goal is to simplify the original polynomial by breaking it down using another polynomial.
This involves:

• Understanding your dividend and divisor.
• Applying the long division technique.
• Carefully tracking each multiplication and subtraction step.
• Attaining a quotient and possible remainder.

This method not only breaks down a polynomial into simpler components, but also helps in understanding the structure and behavior of polynomials. It is a valuable skill in algebra and further provides foundation for more advanced mathematics concepts. Following a step-by-step approach avoids errors and ensures a clear path to the solution.