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Mr. Breitsprecher’s Edition March 9, 2005 Web: www.clubtnt.org/my_algebra
Examples:
Quotient of a Monomial by a Easy does it! Let's look at how • (35x³ )/(7x) = 5x² to divide polynomials by starting Monomial • (16x² y² )/(8xy² ) = 2x with the simplest case, dividing a To divide a monomial by a monomial by a monomial. This is monomial, divide numerical Remember: Any nonzero number really just an application of the coefficient by numerical coefficient. divided by itself is one (y²/y² = 1), Quotient Rule that was covered Divide powers of same variable and any nonzero number to the zero when we reviewed exponents. using the Quotient Rule of power is defined to be one (Zero Next, we'll look at dividing a Exponent Rule). Exponents – when dividing polynomial by a monomial. Lastly, exponentials with the same base we we will see how the same concepts • 42x/(7x³ ) = 6/x² subtract the exponent on the are used to divide a polynomial by a denominator from the exponent on Remember: The fraction x/x³ polynomial. Are you ready? Let’s the numerator to obtain the exponent simplifies to 1/x². The Negative begin! on the answer. Exponent Rule says that any nonzero number to a negative Review: Exponents and Polynomials exponent is defined to be one divided by the nonzero number to n 2 a means the product of n factors, each of which is a (i.e. 3 = 3*3, or 9) the positive exponent obtained. We -2 Exponent Rules could write that answer as 6x .
• Product Role: am*an=am+n Quotient of a Polynomial by a • Power Rule: (am)n=amn Monomial n n n • Power of a Product Rule: (ab) = a b To divide a polynomial by a n n n • Power of a Quotient Rule: (a/b) =a /b monomial, divide each of the terms • Quotient Rule: am/an=am-n of the polynomial by a monomial. • Zero Exponent: a0=1, a ≠0 This is really an example of how we
Working with Polynomials add fractions, recall:
• Adding Polynomials. Remove parenthesis and combine like terms. a + b a b = + • Subtracting Two Polynomials. Change the signs of the terms in the second c c c polynomial, remove parenthesis, and combine like terms. • Multiply Polynomials. Multiply EACH term of one polynomial by EACH Think of dividing a polynomial by a term of the other polynomial, and then combine like terms. monomial as rewriting each term of
Special Products the polynomial over the denominator, just like we did when • FOIL When multiplying binomials, multiply EACH term in the first working with fractions. Then, we binomial by EACH term in the second binomial. This multiplication of simplify each term. terms results in the pattern of First, Outside, Inside, and Last. • Difference Of Squares: (a+b)(a-b) = a2-b2 Example: • Perfect Square Trinomials: