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Chapter 3: Problem 11

Divide using long division. State the quotient, q(x), and the remainder, r(x). $$\frac{4 x^{4}-4 x^{2}+6 x}{x-4}$$

Short Answer

Expert verified
The quotient \(q(x)\) and the remainder \(r(x)\) after performing the long division are \(q(x) = 4x^3 + 16x^2 + 64x +256\) and \(r(x) = 1024\), respectively.

Step by step solution

01

Write the polynomials in the long division format

First, write out the division, putting the dividend \(4 x^{4}-4 x^{2}+6 x\) inside the division symbol, and the divisor \(x - 4\) on the outside. Also, include all missing terms of powers of x in the dividend. If there is no term, write it as 0. Hence, in the dividend write \(4x^{4} + 0x^{3} - 4x^{2} + 6x + 0\).
02

Divide leading terms

Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient \(q(x)\). So \(4x^4梅x = 4x^3\). Write this above the long division line, aligned with \(4x^{4}\).
03

Multiply and subtract

Next, multiply the divisor \(x-4\) with the obtained quotient term \(4x^3\) and subtract it from the dividend. The result is a polynomial of lower degree. This subtracted polynomial becomes the new dividend. Perform the multiplication and subtraction. Thus, \((x-4) \cdot 4x^3 = 4x^4 - 16x^3\). Subtraction gives \(16x^3 -4x^2 + 6x\).
04

Repeat steps 2 and 3 until the degree of the remainder is less than the divisor

Repeat steps 2 and 3. Divide the leading term of this new dividend by the leading term in the divisor. \(16x^3 梅 x = 16x^2\). This is the second term of the quotient. Now multiply \(x-4\) by \(16x^2\) and subtract from the new dividend to get the next dividend. Do this until the degree of the remainder is less than that of the divisor. Thus, after a number of operations, the final quotient and remainder are \(q(x) = 4x^3 + 16x^2 + 64x +256\) and \(r(x) = 1024\).
05

State the quotient and remainder

Finally, state the quotient and the remainder. The quotient is \(q(x) = 4x^3 + 16x^2 + 64x +256\) and the remainder is \(r(x) = 1024\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dividing Polynomials
Polynomial division, much like numerical division, involves dividing a polynomial (the dividend) by another polynomial (the divisor). The goal is to find out how many times the divisor can be multiplied to produce the dividend, a process which provides us with a quotient and possibly a remainder.

When a polynomial is divided by a binomial, the result is a simpler expression and sometimes a remainder. This remainder is a polynomial with a degree less than that of the divisor or is zero. To carry out this division, it's best to organize the terms in descending powers of x and to include any missing terms as zeroes, ensuring that every power of x from the highest down to the constant term is represented.
Long Division Method
The process of polynomial long division follows steps similar to long division with numbers. The steps include dividing the leading terms, multiplying the divisor by the resulting quotient term, and subtracting this product from the dividend to yield a new polynomial. This process is repeated, working with this new polynomial, until the remainder polynomial is of a lower degree than the divisor.

The key is alignment: each step should be clearly marked, and every term, including zeroes for missing powers of x, should align correctly. This helps in visualizing which terms are being manipulated as you divide and subtract.
Quotient and Remainder
Once you finish the long division process, you end up with a quotient and a remainder. The quotient is the polynomial that represents the number of times the divisor fits into the dividend. The remainder, on the other hand, is what's left over and cannot be divided further without going into fractions, as its degree is less than the divisor's.

In our example, the quotient is a cubic polynomial, and the remainder is a constant term, which means we've divided until we could no longer continue without going into decimals or fractions. The result of the division can be expressed as the dividend equals the divisor times the quotient, plus the remainder.

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Most popular questions from this chapter

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