A24 APPENDIX Review Although we have been using x to represent the variable, letters such as y and z are also commonly used. • x x 3 2 4 2 − + is a polynomial (in x ) of degree 4. • y y y 9 2 3 3 2 − + − is a polynomial (in y ) of degree 3. • z5 π + is a polynomial (in z ) of degree 5. Algebraic expressions such as x x x 1 and 1 5 2 + + are not polynomials. The first is not a polynomial because x x 1 1 = − has an exponent that is not a nonnegative integer.The second expression is not a polynomial because the quotient cannot be simplified to a sum of monomials. Now Work problem 25 3 Know Formulas for Special Products Certain products, which we call special products , occur frequently in algebra. We can calculate them easily using the FOIL ( F irst, O uter, I nner, L ast) method of multiplying two binomials. ( )( ) ( ) + + = ⋅ + ⋅ + ⋅ + ⋅ = + + + = + + + First Outer Inner Last ax bcx d axcx axd bcx bd acx adx bcx bd acx ad bc x bd 2 2 Outer First Inner Last Using FOIL (a) x x x x x x 3 3 3 3 9 9 2 2 ( )( ) − + = + − − = − F O I L (b) x x x x x x x x 2 2 2 2 2 4 4 4 2 2 2 ( ) ( )( ) + =+ +=+++=++ (c) x x x x x x x x 3 3 3 3 3 9 6 9 2 2 2 ( ) ( )( ) − =− −=−−+=−+ (d) x x x x x x x 3 1 3 3 4 3 2 2 ( )( ) + + = + + + = + + (e) x x x x x x x 2 1 3 4 6 8 3 4 6 11 4 2 2 ( )( ) + + = + + + = + + EXAMPLE 4 Notice the factors in part (a). The first binomial is a difference and the second one is a sum. Now notice that the outer product O and the inner product I are additive inverses; their sum is zero. So the product is a difference of two squares. Now Work problem 45 Some products have been given special names because of their form.The special products in equations (2), (3a), and (3b) are based on Examples 4(a), (b), and (c). Difference of Two Squares x a x a x a 2 2 ( )( ) − + = − (2)
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