STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS


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1 STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an important first step to begin the study of precalculus Simplifying polynomials A (simple) term is the product of real numbers and variables (with nonnegative exponents) The numerical factor is the coefficient of the term The degree of the term is the sum of the exponents which have variable bases A polynomial is a single term or the indicated sum of two or more terms The operations involving polynomials are illustrated below EXAMPLES: (see Chapter, pages 0 ), Simplify: (x  y)  x( y  x) x  y x  xy x x  y  xy x Multiply: (x )(x  ) x  0x  Recall that the method is: multiply the first terms of each binomial add (mentally if possible) the products of the outside and inside terms multiply the second terms in each binomial A special case of multiplication is squaring a binomial Squaring should be practiced until the product can be written down with no intermediate steps (unless the coefficients are large) (x ) 9x 0x where 9x (x) ; 0x (x)(); () The trinomial 9x 0x is called a perfect square trinomial and can be recognized as such because the first and third terms are perfect squares and the middle term is double the product of x and Simplify: a b  ab 9a b ab ab  a b Factoring Polynomials Factoring means to write a polynomial as a product of expressions The goal is to factor completely so that each factor is prime, that is, each factor cannot be factored further Always look first for a common monomial factor Then try the other factoring possibilities listed below i) Common monomial factoring (ALWAYS CHECK FOR A COMMON FACTOR FIRST!) ii) Trinomial factoring (trial and error method) iii) Difference of two squares: a  b (a b)(a  b) iv) Difference of two cubes: a  b (a  b)(a ab b ) v) Sum of two cubes: a b (a b)(a  ab b ) (The last three formulas above can be checked by multiplication) page
2 EXAMPLES : (see Chapter, pages  ) Factor completely 0x y  xy  y y( 0x  x  ) common factor of y y (?x )(?x ) possibilities are x & 0x; x & 0x: x & x y(x?)(x?) possibilities are: & ;  & ; & ;  & y(x )(x  checked by multiplication a  (a  )(a a 9) the difference of two cubes, see formula iv) above a b  9ab ab(a  9b ) common factor of ab ab(a b)(a  b) using the difference of two squares formula The Factor of (): It is always possible to factor out a () from any polynomial, resulting in every sign of the polynomial being reversed If the polynomial is a binomial such as x  y then x  y ()(x y) ()(y  x) Thus, if two terms are being subtracted, factoring out () merely reverses the order of the terms of the binomial EXAMPLES: a) x y x x(xy ) OR (x)(xy  ) b) a  b ()(b  a) Factoring by Grouping: (See Chapter, page ) Grouping may work to factor a polynomial of or more terms The goal is to group terms to create a common factor, a difference of two squares or some other familiar factoring form EXAMPLES: 9 a  x  x  a  (x x ) a  (x ) difference of two squares [a (x )][a  (x )] (a x )(a  x  ) 0 x  0x  9x x (x  )  (x  ) common factor of (x  ) (x  )(x  ) A Least Common Multiple (LCM) is the quantity used: ) as a lowest common denominator (LCD) when adding fractions; ) to clear fractions occurring in equations and some inequalities; ) to simplify complex fractions To find the LCM first factor each polynomial Then for each distinct factor pick the largest exponent (See Chapter, page ) EXAMPLES: Find the LCM of the following: x  9 (x )(x  ) The LCM is: (x )(x  )(x ) x x  (x  )(x ) x x (x x ) (x ) page
3 Polynomial Fractions or Rational Expressions (See Chapter, pages  9) Factoring is a key skill when working with rational expressions EXAMPLES Reduce: x  0x x first factor numerator and denominator completely () (x  x  ) (x  )(x ) ( x)(  x) ( x)(  x) cancel any like factors () Note that (x  ) ()(  x)! (x ) x A fraction such as a   a can always be reduced by factoring out a (), namely, a   a ()(  a)  a  Equivalently, canceling leaves a factor of () in the numerator and of in the denominator Divide:  x x  x  x  9 x  Invert the divisor and multiply  x x  x  x  x  9 Factor top and bottom completely (  x) (x  ) (x )(x  ) (x )(x  ) () () (  x) (x  ) (x )(x  ) (x )(x  ) () () cancel any like factors  (x )(x ) Add (or subtract): a  a   a a   a Find the LCD  a (a  )  a ( a)(  a) Do not include both (a  ) & (  a) in the LCD (a  ) a ( a)(  a)  (  a)  a ( a)(  a) LCD: (  a)( a)  (  a) a a  a ( a)(  a) ( a) (  a) (  a)( a)   a  a (  a)( a)  a (  a)( a) page
4 COMPLEX FRACTIONS (See Chapter, pages 90) A complex fraction is a fraction that has fractions in the numerator or in the denominator or both Complex fractions will occur frequently throughout this course A recommended method of simplifying is to multiply by a carefully selected name for one, that is, simplify by multiplying top and bottom by the same expression EXAMPLES: Simplify:  The, & are the "extra" denominators of the complex fraction The LCM of, & is Multiply by to simplify this complex fraction With practice the second step above can be eliminated Simplify: x  x  (x ) (x )  (x ) (x )(x  )  x  (x )(x  )  x (x )(x  ) (  x) (x )(x  )  (x ) OR (x ) Simplifying Radicals: (See Chapter, pages  ) The radicand of 9 is 9 and the order of the radical is Simplifying radicals means to change to an equivalent form: i) with no perfect nth power factors in the radicand (order n radicals), and ii) with an integer radicand (or sometimes no radical in the denominator) EXAMPLES: Simplify a) 9 9 b) Remember to cancel like factors before multiplying! page
5 a) 0 0 OR 0 b) Operations with Radicals: (See Chapter, pages  ) Radicals with the same radicand and same order are like radicals Add or subtract coefficients of like radicals Radicals of the same order can be multiplied by multiplying the radicands Similarly, radicals of the same order can be divided by dividing the radicands EXAMPLES: Perform the indicated operations 9 First simplify each radical 0 ( ) (  )   (Similar to polynomial multiplication!)   If the denominator has two or more terms, the simplified form will be an equivalent form with a rational number denominator (actually the denominator will become an integer) Multiply top and bottom by the conjugate to simplify Note that (  ) ( ) has the form of multiplying (a b)(a  b) a  b Imaginary or Complex Numbers (See Chapter, pages  ) Expressions such as  do not represent real numbers since no real number squared equals  In this case,  9 () i where i , an imaginary number page
6 EXAMPLES: Simplify i  i (  i) (  i) OR  i Radicals and Exponents (See Chapter, pages  ) Recall that: b 0, b 0;  ; / ; / Also, for x 0, x a x b x ab ( xy ) x a x b x ab x y ( x a ) b x ab a a x a y a xa y a As noted previously it is not possible to multiply because the orders are different However, using exponential forms sometimes allows some simplifying EXAMPLES: Perform the indicated operations / / // / or or  (  9 / ) 9 / xy x  y  x y xy  x xy y Note that this was merely a complex fraction! y x x y Simplify and write the answer without negative or zero exponents x / y  / / x / y  / Simplify inside the parentheses first  /  (/) Subtract exponents of like bases: x // y (Note:   AND / 9  ) So, x / y  / x / y  / / x / y  / / / x / / y / / page
7 x / x / y / y / Linear Equations and Inequalities (See Chapter, pages , Chapter, page ) EXAMPLES: Solve x x  0 C lear fractions by multiplying each side by 0, the LCM of the denominators x 0 0 x  x 90x  Get all terms with the variable on one side and everything else on the other side 90x  x x So, x and the solution set is S 9 x Recall that statements of the form a x b are implied "AND" statements and mean that a x AND AT THE SAME TIME x b It is shortest to solve both at once as shown below x  x Clear fractions first (unless the denominators contain variables!) Add  to each part of the inequality to isolate the x term 9 x Divide each part by to solve for x 9 x 9 x IF YOU MULTIPLY OR DIVIDE AN INEQUALITY BY A NEGATIVE NUMBER, REMEMBER THAT THE INEQUALITY REVERSES! 0 Solve for c: abc bc  ab Isolate all terms that contain "c" ab bc  abc Factor out "c' from the right side ab c(b  ab) Divide each side by the coefficient of "c" ab ab a c b  ab b(  a)  a Completing the Square (See Chapter, page ) In addition to deriving the quadratic formula from ax bx c 0, a 0, completing the square has other important uses which will be encountered this semester as well as in future courses x   x : To complete the square, add  9 page
8 So, x  x 9 x  Quadratic Equations (See Chapter, pages  0) Quadratic equations have the form ax bx c 0, a 0 EXAMPLES: Solve (x )(x  ) x  9 Remove parentheses; x  x  0 x  9 x  x  0 Get a zero on one side Factor the nonzero side (x )(x  ) 0 Set each factor containing a variable equal to zero x 0 OR x  0 x OR x The solution set S, If the nonzero side cannot be factored using integers, then use the quadratic formula x b ± b  ac a and simplify x  x  0 The left side does not factor Use the quadratic formula with a, b , c  x () ± ()  ()() () ± ( ± ) ± ± ± Equations of the form x k for any real number k A special case of quadratic equations occurs when b 0, that is, the quadratic equation has the form x k for any real number k The most efficient method for solving these equations is to use the following x k then x ± k Notice that the square root is only on the right side!! In fact, x x for all real numbers x! x Special case of a quadratic equation x Solve for x x ± ± ± ± OR ± page
9 Equations with rational or radical expressions (See Chapter,  ) The goal is to convert to an equivalent equation that is either a familiar linear or quadratic form EXAMPLES: Solve x x(x ) x The LCM of the denominators is x(x ), x 0,  x(x ) x x(x ) x Clear fractions x x A linear equation; solve for x x  However, x  makes the LCM and a denominator equal to zero So, x  cannot be a solution and x  is called an extraneous root The solution set is the empty set, namely, S x  x Isolate the radical; if there are two radicals, isolate one of them x   x Square each side This could result in extra solutions! ( x  ) (  x ) x x x A quadratic equation 0 x  x (x  )(x  ) x OR x are POSSIBILITIES ONLY Each must be checked using the original equation Check x : Check x : Does ? Does ? 9 x is NOT a solution x is a solution The solution set is S {} page 9
10 Systems of Two Equations (See Chapter, pages  ) Systems of two or more equations can be solved by the Addition (Elimination) Method or by Substitution Each method is illustrated below EXAMPLES: Solve Use the Addition or Elimination Method x y x  y 0 Multiply the first equation by and the second equation by x y 9x  y 0 Add the equations to eliminate the y terms x x Substitute into either equation to find y Using the first equation () y y  y  The solution is the ordered pair (, ) The above system could be solved by substitution as well Use the Substitution Method y  x y x Solve the first equation for y and substitute into the second equation y x (x ) x, a quadratic equation in one variable x x 0 x  x  (x )(x  ) So, x  OR x Find the corresponding y value for each x value x  y  So, one solution is the ordered pair (, ) x y So, another solution is the ordered pair (, ) The solution set S {(, ), (, )} page 0
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