Appendix A.3Polynomials and FactoringA27PolynomialsThe most common type of algebraic expression is the polynomial.Some examples areand The first two are polynomials inand the third is a polynomial inandThe terms of a polynomial in have the formwhere is the coefficientand is the degreeof the term. For instance, the polynomialhas coefficients 2,0, and 1.Polynomials with one, two, and three terms are called monomials, binomials,andtrinomials,respectively. In standard form,a polynomial is written with descendingpowers of Writing Polynomials in Standard FormLeadingPolynomialStandard FormDegreeCoefficienta.7b.2c.8808Now try Exercise 19.A polynomial that has all zero coefficients is called the zero polynomial,denotedby 0. No degree is assigned to this particular polynomial. For polynomials in more thanone variable, the degree of a termis the sum of the exponents of the variables in theterm. The degree of the polynomialis the highest degree of its terms. For instance, thedegree of the polynomial is 11 because the sum of the exponentsin the last term is the greatest. The leading coefficient of the polynomial is the coefficient of the highest-degree term. Expressions are not polynomials if a variable isunderneath a radical or if a polynomial expression (with degree greater than 0) is in thedenominator of a term. The following expressions are not polynomials.The exponent “” is not an integer.The exponent “” is not a nonnegative integer.1x25xx25x11 2x33xx33x1 22x3y64xyx7y488x099x2449x255x74x23x24x25x723xExample 1x.5,2x35x212x35x20x1kaaxk,xy.xx5x2y2xy3.3x47x22x4,2x5,A.3POLYNOMIALS ANDFACTORINGWhat you should learn•Write polynomials in standard form.•Add, subtract, and multiply polynomials.•Use special products to multiplypolynomials.•Remove common factors from polynomials.•Factor special polynomial forms.•Factor trinomials as the product oftwo binomials.•Factor polynomials by grouping.Why you should learn itPolynomials can be used to modeland solve real-life problems. Forinstance, in Exercise 224 on page A37, a polynomial is used to model the volume of a shipping box.Definition of a Polynomial in xLet be real numbers and let be a nonnegative integer. A polynomial in is an expression of the formwhere The polynomial is of degreeis the leading coefficient,andis the constant term.a0ann,an0.anxnan1xn1 . . .a1xa0xna0, a1, a2, . . . , an
A28Appendix AReview of Fundamental Concepts of AlgebraOperations with PolynomialsYou can add and subtract polynomials in much the same way you add and subtract realnumbers. Simply add or subtract the like terms(terms having the same variables to thesame powers) by adding their coefficients. For instance,and are like termsand their sum isSums and Differences of Polynomialsa.