Problem 6 - Basic Technical Mathematics with Calculus

Question

Find the remainder by long division. $$\left(x^{4}-4 x^{3}-x^{2}+x-100\right) \div(x+3)$$

Step-by-Step Solution

Answer

The remainder is 77.

Step 1: Divide First Term

Divide the first term of the dividend, \(x^4\), by the first term of the divisor, \(x\): \(x^4 \div x = x^3\). Write \(x^3\) as the first term of the quotient.

Step 2: Multiply and Subtract

Multiply \(x^3\) by the divisor \((x+3)\) to get \(x^4 + 3x^3\). Subtract \(x^4 + 3x^3\) from the current dividend, resulting in \(-7x^3 - x^2 + x - 100\).

Step 3: Repeat Division

Divide the new leading term \(-7x^3\) by \(x\), resulting in \(-7x^2\). Add \(-7x^2\) to the quotient.

Step 4: Multiply and Subtract Again

Multiply \(-7x^2\) by \(x+3\) to obtain \(-7x^3 - 21x^2\). Subtract this from the current dividend to get \(20x^2 + x - 100\).

Step 5: Continue with Division

Divide \(20x^2\) by \(x\) to get \(20x\). Add \(20x\) to the quotient.

Step 6: Multiply and Subtract Once More

Multiply \(20x\) by \(x+3\) to obtain \(20x^2 + 60x\). Subtract from the previous result to get \(-59x - 100\).

Step 7: Final Division Step

Divide \(-59x\) by \(x\) to get \(-59\). Add \(-59\) to the quotient.

Step 8: Final Multiply and Subtract

Multiply \(-59\) by \(x+3\) to get \(-59x - 177\). Subtract from \(-59x - 100\) to find the remainder, which is \(77\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a handy tool in algebra that connects polynomial division and evaluation. It states that when a polynomial \( f(x) \) is divided by a linear divisor \( x-c \), the remainder of the division is equal to \( f(c) \). This theorem can simplify the process when you're only interested in finding the remainder, without going through the entire division process. For example, in the given exercise, if you wanted to quickly find the remainder of dividing \( x^4 - 4x^3 - x^2 + x - 100 \) by \( x+3 \), you can substitute \(-3\) into the polynomial and evaluate. This saves time and effort compared to performing the entire polynomial long division.

Algebraic Division

Algebraic division refers to the method of dividing polynomials, similar to how we handle numbers in arithmetic division. It involves repeatedly dividing, multiplying, and subtracting terms.
In the problem provided, we divided the polynomial \( x^4 - 4x^3 - x^2 + x - 100 \) by \( x+3 \) using steps analogous to long division with numbers. Each part of the dividend is divided by the leading term of the divisor, \( x \), and the result is multiplied back against the entire divisor. The product is then subtracted from the dividend, and the process repeats with the new polynomial obtained after subtraction. This repetitive cycle continues until the degree of the remainder is less than the degree of the divisor.

Polynomial Division Steps

The polynomial long division involves a clear, step-by-step process similar to traditional long division used in arithmetic. Here's a breakdown:

• **Step 1:** Divide the highest degree term of your polynomial (dividend) by the highest degree term of the divisor. This gives the first term of your quotient.
• **Step 2:** Multiply the entire divisor by the term you found in Step 1, then subtract the result from your original polynomial.
• **Step 3:** Bring down the next term from the original polynomial, if necessary, and repeat the process with the new polynomial.
• **Step 4:** Continue these steps until the remainder's degree is lower than the divisor's. In this exercise, each step eventually leads to finding the quotient and the final remainder of 77, which signifies the part left after division.

The polynomial division steps are crucial, as they systematically reduce the polynomial, ensuring a clear and correct solution. Understanding each phase helps demystify the process, making algebraic division feel more straightforward.