Mid-Chapter Quiz
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Exercise 24 Page 690
Recall the definition of a monomial and a polynomial.
See solution.
Practice makes perfect
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)Ă· 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^2(9x^3-6x^2+15x+3)*1/3x^2
Distr
Distribute 1/3x^2
9x^3*1/3x^2-6x^2*1/3x^2+15x*1/3x^2+3*1/3x^2
MoveLeftFacToNumOne
a* 1/b= a/b
9x^3/3x^2-6x^2/3x^2+15x/3x^2+3/3x^2
ReduceFrac
a/b=.a /3./.b /3.
3x^3/x^2-2x^2/x^2+5x/x^2+1/x^2
DivPow
a^m/a^n= a^(m-n)
3x^(3-2)-2x^(2-2)+5x/x^2+1/x^2
â–Ľ
Simplify
SubTerms
Subtract terms
3x^1-2x^0+5x/x^2+1/x^2
ExponentOne
a^1=a
3x-2x^0+5x/x^2+1/x^2
ExponentZero
a^0=1
3x-2+5x/x^2+1/x^2
ReduceFrac
a/b=.a /x./.b /x.
3x-2+5/x+1/x^2
polynomial partof 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.
5/x+1/x^2
â–Ľ
Rewrite
ExpandFrac
a/b=a * 3x/b * 3x
15x/3x^2+1/x^2
ExpandFrac
a/b=a * 3/b * 3
15x/3x^2+3/3x^2
AddFrac
Add fractions
15x+3/3x^2
Polynomial Long Division
Let's recall the steps of polynomial long division.
- 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.
- Divide the firstterm of the dividend by the first term of the divisor. The result is the first term of the quotient.
- Multiply the first term of the quotient by the divisor and write the product under the dividend.
- Subtract the product from the dividend. We will call the result of this operation the remainder.
- 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.
l r 3x^2 & |l 9x^3-6x^2+15x+3
â–Ľ
Divide
9x^3/3x^2= 3x
r 3x r 3x^2 & |l 9x^3-6x^2+15x+3
Multiply term by divisor
r 3x rl 3x^2 & |l 9x^3-6x^2+15x+3 & 9x^3
Subtract down
r 3x r 3x^2 & |l - 6x^2+15x+3
â–Ľ
Divide
- 6x^2/3x^2= - 2
r 3x-2 r 3x^2 & |l - 6x^2+15x+3
Multiply term by divisor
r3x - 2 rl 3x^2 & |l - 6x^2+15x+3 & - 6x^2
Subtract down
r 3x-2 r 3x^2 & |l 15x+3
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.
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