Exercise 24 Page 690

Let's start by recalling the definitions of two important concepts — monomial and polynomial.

• Monomial: a real number, a variable, or a product of a real number and one or more variables with whole number exponents.
• Polynomial: a monomial or a sum of monomials.

When dividing a polynomial by a monomial, we can take two different approaches: multiply the polynomial by the reciprocal of the monomial or use polynomial long division. Let's review these two methods using the same example. (9x^3-6x^2+15x+3)÷ 3x^2

Multiplying by the Reciprocal

We will divide our polynomial by the monomial. The first method we use will be multiplying the polynomial by the reciprocal of the monomial. (9x^3-6x^2+15x+3)÷ 3x^2 ⇕ (9x^3-6x^2+15x+3)*1/3x^2Let's simplify the above expression using the Distributive Property.

The result is 3x-2+ 5x+ 1x^2. Note that this is not a polynomial because the last two terms have variables whose exponents are not whole numbers. 3x-2+5/x+1/x^2 ⇔ 3x-2+5x^(- 1)+x^(- 2) However, when dividing polynomials we usually state our answer as a quotient and a remainder which should both be polynomials. The quotient is the polynomial part of the obtained expression. Quotient: 3x-2 To identify the remainder we should rewrite the remaining terms as a rational expression with a denominator equal to the divisor.

The remainder is the numerator of the above rational expression. Remainder: 15x+3 Now that we have the remainder, we can write the quotient of the expression in two different ways. 3x-2 + 15x+3/3x^2 or 3x-2 R15x+3 Another way of identifying the remainder is to not simplify the terms of the expression obtained after using Distributive Property if their numerator has a lower degree than their denominator.

Polynomial Long Division

Let's recall the steps of polynomial long division.

1. Write the dividend and the divisor in standard form. If a term is not present in the dividend, add the term with a coefficient of 0.
2. Divide the firstterm of the dividend by the first term of the divisor. The result is the first term of the quotient.
3. Multiply the first term of the quotient by the divisor and write the product under the dividend.
4. Subtract the product from the dividend. We will call the result of this operation the remainder.
5. Repeat Steps 2 through 4 treating the remainder as the dividend. Stop when the degree of the remainder is less than the degree of the divisor.

The terms of our dividend are in descending degree order, so it is already written in standard form. Additionally, it has no missing terms. Since our divisor is a monomial, it is also written in standard form. We can go straight to dividing!

The degree of the remainder is 1 and the degree of the divisor is 2. Therefore, the degree of the remainder is less than the degree of the divisor. We must stop the division right here. We have that the quotient is 3x-2 with a remainder of 15x+3. ( 9x^3-6x^2+15x+3)÷3x^2 = 3x-2+15x+3/3x^2 As previously mentioned, this could also be written as 3x-2 R15x+3. Note that when the divisor is a monomial, each round of Steps 2 through 4 results in removing the first term of the current dividend.

Conclusion

Both methods give the same result but they consist of different steps. Multiplying by reciprocal uses the Distributive Property and the Quotient of Powers Property. Polynomial long division repeats the steps of divide, multiply, and subtract. When the divisor is a monomial, polynomial long division is often a longer process but it is up to you to choose which method you prefer.