Question

Use long division to divide. Divisor \(x^{3}-1\) Dividend $$x^{5}+7$$

Short answer

The quotient is \(x^{2}\) and the remainder after dividing is \(x^{2}+7\).

Step-by-step solution

Setup the long division

To begin the process, you organize the setup similar to regular long division: the divisor \(x^{3}-1\) outside the long division symbol and the dividend \(x^{5}+7\) on the inside.

Begin the division

Observe the highest degree term in the divisor and the dividend. The highest degree term in the divisor is \(x^{3}\) and in the dividend it is \(x^{5}\). The question is: how many times does \(x^{3}\) go into \(x^{5}\)? To get the first term of the quotient, divide \(x^{5}\) by \(x^{3}\) to get \(x^{2}\). Write \(x^{2}\) on top of the division symbol.

Multiply and subtract

Now, multiply the divisor by the term you found in Step 2 (\(x^{2}\)) and subtract this from the dividend: \(x^{5}+7-(x^{2}*(x^{3} - 1)) = x^{5}+7-(x^{5} - x^{2})= x^{2}+7\). Write this new polynomial under the old one and draw a line beneath it. Now your task is to divide \(x^{2}+7\) by \(x^{3}-1\)

Finish the division

Observe that you cannot divide \(x^{2}+7\) by \(x^{3}-1\) as the degree of the divisor is larger than the degree of the dividend left after subtraction, which is the remainder we have after dividing. So, the division stops here.

Key concepts

Dividing Polynomials

Understanding how to divide polynomials is essential in algebra. It's quite similar to long division with numbers, with a focus on matching the highest degree terms. To divide a polynomial by another polynomial, you compare the highest degree terms to find the quotient. Then, you multiply this quotient by the divisor and subtract it from the dividend. This process is repeated until the degree of the remaining polynomial is less than the degree of the divisor, or until no remainder is left.

For the exercise at hand with divisor \(x^{3}-1\) and dividend \(x^{5}+7\), we find that the highest degree term of the dividend is \(x^{5}\) and of the divisor is \(x^{3}\). We then take the ratio of these terms to start our division and proceed as detailed in the solution.

Synthetic Division

Synthetic division is an efficient alternative to polynomial long division, especially when dividing by a linear factor. This quick method involves using only the coefficients of the polynomials. It streamlines the process by removing variables and exponents, setting up a small table where you only perform operations with numbers. This technique is not used in our specific exercise because synthetic division best applies when the divisor is a linear expression (form \(x - c\)), not a cubic polynomial like \(x^{3}-1\).

Remainder Theorem

The Remainder Theorem is a powerful concept to understand when dealing with polynomial division. It states that when a polynomial \(f(x)\) is divided by a linear divisor \(x-c\), the remainder of the division is equal to \(f(c)\). In the case of the given exercise, we're left with a remainder of \(x^{2}+7\) after attempting the long division. As the divisor is \(x^{3}-1\), which isn't linear, this theorem doesn't directly apply, but it's good to be aware of this theorem for solving similar problems with linear divisors.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and non-negative integer exponents. The polynomials involved in our division problem are examples of algebraic expressions. Recognizing the structure of these expressions is necessary to perform polynomial division, as we need to align terms with the same powers before subtracting. The final remainder \(x^{2}+7\) is also an algebraic expression and represents what's left after the division process is completed.