Exercise 25 Page 690

Let's start by recalling a few basic definitions.

Monomial: a product of numbers and variables raised to non-negative powers.
Polynomial: a sum of multiple monomials.
Degree of a Monomial: the sum of powers of variable factors of the monomial.
Degree of a Polynomial: the highest degree of the monomials forming the polynomial.

Keeping all of these definitions in mind, let's move on to our task. Assume we are dividing a polynomial in one variable with a degree of 5 by a monomial in the same variable with a degree of 2. Let x be the variable. We can write the general form of our polynomial and monomial using letters to represent unknown coefficients.

Polynomial: & ax^5 +bx^4 +cx^3 +dx^2 +ex+f Monomial: & gx^2 Here a, b, c, d, e, f, and g represent real numbers. Since the degree of the polynomial is 5, a must be different from 0. Similarly, g≠ 0. Below we have included a few examples of polynomials and monomials that follow these general forms.

Polynomial	Monomial
2x^5	3x^2
3x^5+2x^3-1	x^2
x^5+3x^4-7x^3-x^2+2x+8	- 9x^2

We are interested in determining the degree of the result of the division. (ax^5+bx^4+xc^3+dx^2+ex+f)÷ gx^2 Let's calculate the first term of the quotient as we would do when using polynomial long division. This means we will divide the first term of the dividend by the first term of the divisor. ax^5/gx^2=a/gx^3 When using long division, the first calculated term of the quotient is always the one with the highest degree. Therefore, the degree of the quotient is 3 which is less than 5. Since we used the general form of the polynomial and the monomial, we can expect that the degree of the quotient will always be less than 5.

Extra

Determining the Degree of The Quotient

Knowing the degree of the dividend and the degree of the divisor, we can always determine the degree of the quotient. Assume we divide a polynomial with degree m by a polynomial with degree n. Note that n must be less than or equal to m. Dividend a_mx^m+a_(m-1)x^(m-1)+...+a_1x+a_0 Divisor b_nx^n+b_(n-1)x^(n-1)+...+b_1x+b_0 Remember that a_m≠ 0 and b_n≠ 0. As previously mentioned, we can find the term of the quotient with the highest degree by dividing the first term of the dividend by the first term of the divisor.

a_mx^m/b_nx^n

WriteProdFrac

Write as a product of fractions

a_m/b_n*x^m/x^n

DivPow

a^m/a^n= a^(m-n)

a_m/b_nx^(m-n)

We can see that the degree of the term is m-n and therefore the degree of the quotient is m-n. Since the degree of a polynomial is non-negative, m-n is less than or equal to m. The degree of the quotient m-n is equal to m if and only if n=0. This is only when the divisor is a constant.

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