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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

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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 how 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: (a+b)2 = a2+2ab+b2 • (16x³ - 12x² + 4x)/(2x) (a-b)2 = a2 – 2ab + b2 = (16x³)/(2x)-(12x² )/(2x)+(4x)/(2x) = 8x² - 6x + 2 Mr. Breitsprecher’s Edition March 9, 2005 Algebra Connections, Page 2 Quotient of a Polynomial by a subtrahend and add (add the Polynomial opposite). Online To divide a polynomial by a polynomial, 5. Repeat steps 2 through 4 until use long division, similar to the long the difference you obtain is a Resources division technique used in arithmetic. polynomial of degree less than How to Add, Subtract, Multiply, Remember in starting the long division the degree of the divisor. and Divide Polynomials process: http://faculty.ed.umuc.edu/~swalsh/ 6. If the final difference is zero, 1. Write dividend and divisor in terms the division is exact. The Math%20Articles/Polynomial.html of descending powers of variable, quotient is the polynomial given Dividing Polynomials leaving space for any missing across the top. If the difference powers of the variable or writing in in nonzero division is not exact, http://www.wtamu.edu/academic/an the missing powers with coefficient the quotient is the polynomial ns/mps/math/mathlab/int_algebra/int zero. If there is more than one given across the top plus the _alg_tut35_div.htm variable, arrange dividend and remainder (polynomial in last http://tutorial.math.lamar.edu/AllBr divisor in terms of descending order line) divided by the divisor. owsers/1314/DividingPolynomials.a (from the term with the highest sp Example: “degree” to the lowest “degree”). Dr. Math: Dividing Polynomials. 2. Divide first term of divisor into first 2x 2 + x + 3 R(−1) /(2x −1) http://mathforum.org/library/drmath/ term of dividend (On subsequent 3 2 view/52901.html iterations, into first term of previous 2x +1 4x + 0x + 5x − 4 difference). Place this answer above Long Division of Polynomials. 4x 3 − 2x 2 long division symbol. http://www.sosmath.com/algebra/fa 2 3. Multiply divisor by the expression 2x − x ctor/fac01/fac01.html 6x − 4 just written above division symbol http://www.purplemath.com/module and align like terms. 6x − 3 s/polydiv2.htm 4. Subtract line just written from line −1 http://www.mathwords.com/p/polyn immediately above it. Remember to omial_long_division.htm subtract we change the sign of the Source: http://home.sprynet.com http://www.math.utah.edu/online/10 10/euclid/ http://tutorial.math.lamar.edu/AllBr owsers/1314/DividingPolynomials.a sp Polynomial Division and Factoring One thing you will do when Definitions http://campus.northpark.edu/math/Pr solving polynomials is combine like eCalculus/Algebraic/Polynomial/Fa terms. Understanding this is the key Monomial has one term: 5y or -8x2 or 3. ctoring to accurately working with Binomial has two terms: -3x2 + 2, or 9y polynomials. Let’s look at some Online Solutions. Scroll down this - 2y2 examples: site to the form identified as Division of polynomials: p(x) / Trinomial has 3 terms: -3x2 + 2 +3x, or • Like terms: 6x + 3x - 3x f(x). It will divide polynomials to 2 9y - 2y + y • NOT like terms: 6xy + 2x - 4 degree 6. Just set the coefficients, and click "Divide". Be sure that the Degree Of The Term is the exponent of The first two terms are like and they degree of p(x) >= degree of f(x). the variable: 3x2 has a degree of 2. can be combined: Leading zero coefficients are ignored – use zeros for terms in the Degree of a Polynomial is the highest 5x2 + 2x2 – 3 degree of any of its terms. form that are not present in the Combining like terms, we get: problem you want to solve. When the variable does not have an 7x2 – 3 http://www.egwald.com/linearalgebr exponent -understand that there's a '1'. a/polynomials.php FREE Tutoring And Academic Support Services!!! Basement of McCutchan Hall, Rm. 1 Mon-Thurs: 9 a.m. – 9 p.m. Academic Support Services Fri: 9 a.m. – 3 p.m. and Sun 5 p.m. – 9 p.m. .
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