Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE


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1 Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers with the same base. o Eample: = a a m n = a mn : Subtract eponents when dividing powers with the same base. 9 Eample: o (ab) m = a m b m : Multiply eponents when raising a power to a power. Eample: ( ) = 1 Evaluate an epression with a number raised to a 0 power it is 1. o Eample: 0 = Evaluate epressions with negative eponents: a n = a n or a n = n, a 0 a 1 1 z o Eamples:  = ; =  ; y  z = y Evaluate epressions using scientific notation: c 10 n, where 1 c < 10 and n is an integer o Eamples: 1,00 = ; 0.00 = Multiply/divide numbers in scientific notation by using the laws of eponents; you may need to put the number back into scientific notation! o Multiply or divide coefficients; multiply or divide eponents.0 10 o Eamples: (.110 )(.10  ) = 9.10 ; = Add and Subtract Polynomials o A monomial is a number, variable, or product of a number and one or more variables with a whole number coefficient. Since variables in a denominator represent negative coefficients in the numerator, monomials do not have variables in the denominator. o Eample:, 1 y are monomials;, are not o The degree of a monomial is the sum of the eponents of the variables in the monomial. Eample: the degree of y is The degree of a constant (such as ) is 0 o A polynomial is a monomial or the sum of monomials. o By convention, we write the terms of a polynomial in degree order, from greatest to least (standard form). o The degree of a polynomial is the greatest degree of its terms, that is, it takes the degree of the monomial with the largest degree in the polynomial.
2 Algebra 1 Laws of Eponents/Polynomials Test Study Guide Page Eample: the degree of y + + y + is o Add polynomials by combining like terms. o Subtract polynomials by distributing the negative sign, and then adding. Multiplying Polynomials o Multiply a monomial by a polynomial by distributing the monomial over all the terms in the polynomial. o Multiply polynomials by performing multiple distribution for each term in the first polynomial over every term in the second monomial. o The FOIL acronym ( First, Outside, Inside, Last ) is a way to remember how to multiply two binomials. Special Polynomial Products o Square of a polynomial pattern: (a + b) = a + ab + b and (a b) = a ab + b. o Sum and difference pattern: (a + b)(a b) = a b Solving Polynomials o Make sure all terms are one side of the equation, and 0 is on the other. o Factor completely (see below for factoring summary). o Use the zero product property (if ab = 0, then a = 0 or b = 0) to solve. Factoring o Always apply Type I factoring (factor out GCF) before factoring any polynomial!! o Always multiply your answer back to a polynomial to verify!! o Type I Factoring factor out GCF o Factor out the Greatest Common Factor (GCF) of the terms in the polynomial o Eample: + = ( + ) o Type II Factoring notice special patterns o If polynomial follows special product pattern (as described in section 9.), we can easily factor: a b = (a + b)(a b) or a + ab + b = (a + b) o Eample: 1 = ( + 9)( 9) o Type III Factoring Factor + b + c o Draw two sets of parentheses, with the variable as the first term in each. o Find two factors of c that add up to b (when c is positive) or subtract to b (when c is negative) o When c is positive, the signs in the parentheses will be the same (the sign of b) o When c is negative, the signs in the parentheses will be different (the bigger number takes the sign of b) o Eample:  = ( )( + ) o Type IV Factoring Factor by Grouping o Applies when a polynomial has terms o Group the first two terms, and the second two terms; factor out the GCF for each set o Factor out the common polynomial o Eample: + = ( ) + ( ) = ( ) + ( ) = ( )( + ) o Type V Factoring Factor a + b + c
3 Algebra 1 Laws of Eponents/Polynomials Test Study Guide Page o There are several methods for factoring here is the grouping method: Multiply ac, and find the factors of ac that add or subtract to b. Rewrite the polynomial, splitting up b into the sum found above. Factor by grouping. o Eample: Factor + a c = = ; the factors of that add up to are 1 and Rewrite:  + Group and factor: ( ) + ( + ) = ( )  ( ) Factor common binomial: ( )( 1) o A polynomial is factored completely when each factor is prime, that is, each factor cannot be factored further. Always check each factor for further factoring (especially look for further Type I (GCF) or Type II (special product) factoring). Study Questions 1) Evaluate the epressions, writing answers using positive eponents: a) 11 b) (1 ) 10 c) ( 1) d) e) f) g) 1 11 h) i) 1 j) ( y ) k) (y ) y l) b c m) n) 1 w 1 o) 1 1 p) m n 10 q) 10 y z 1 0 r)  s) t) ()  1 u) 010 v) (y) w) ( y  )  ) ) Rewrite the numbers in scientific notation. a),100 b),,000 c) 0.0 d) ) Rewrite the numbers in standard form. a).0 10 b).1 10 c) d). 101
4 Algebra 1 Laws of Eponents/Polynomials Test Study Guide Page ) Multiply or divide. Epress results in scientific notation. a) (.9 10 )(. 10 ) b) (. 10 )( ) c) d) ) Tell whether the following are monomials; if yes, what is the degree; if not, why not? 1 a) y z b) c) y ) What is the degree of the polynomial? y + y ) Add or subtract the polynomials: a) (a ) + (a 1) b) (m m + ) + (m + 10m + ) c) (9b 1b + b) (1b  b + 1) ) Multiply the polynomials: a) ( + 9) b) b (b b + b 11) c) (b )(b b + 1) d) (y + )(y ) e) ( + )( ) f) ( + ) g) ( )( ) h) (w + )(11w ) i) (a )(a ) 9) Divide the polynomials: a) b) c)
5 Algebra 1 Laws of Eponents/Polynomials Test Study Guide Page 10) Factor completely: a) + y b) w w c) d) e) n n + f) y  y + 1 g)  h) 10 + i) ) Solve (find the roots): a) ( )( + ) = 0 b) + = 1 c) = 0 d) 10 + = 0 e) = 0 f) m m = m  1 g) a + a = 0 h) + 0 = 0 1) Find the zeros of the functions. a) f() = +  b) f() = c) f() =  1 1) The length of a rectangle is inches more than times its width. The area of the rectangle is square inches. What is the width? +
6 Algebra 1 Laws of Eponents/Polynomials Test Study Guide Page STUDY QUESTION ANSWERS 1) a) 1 b) 1 0 c) 1 d) 1 e) f) g) 1 h) i) 1 b 1 m j) 9 y 10 k) y 1 l) m) n) o) 10 p) 0 c w n y q) 1 r) s) t) u) undefined v) 1y w) )  0 ) a).1 10 b). 10 c) d) ) a) 0,000 b),1 c) d) 0. ) a) b) c).0 10 d).10  ) a) yes; degree is 9 b) no; contains eponents that are not whole numbers c) no; can t have variables under the radical sign ) degree is ) a) 1a b) m + 9m + 9 c) 9b + b 1 ) a) + 9 b) 0b + 10b b + b c) b  b + b  d) y + y 0 e) 1 f) g) h) w + w 1 i) 1a a + 9) a) + b) + c) ) a) ( + y) b) w (w ) c) ( + )( ) d) ( + 9)( + ) e) (n )(n  ) f) (y  )(y + ) g) ( )( + ) h) ( 1)( ) i) ( + 1)( + 1)( 1) 11) a) {, } b) {, } c) {, 9} d) {) e) {, } f) {,, } g) {, 0} h) {0, 1, } 1) a) =1, 1) in. b) = 0,, c) = ,
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