xyz Degree is 5. See last term.

Transcription

1 THE PERFECT SQUARE - COLLEGE ALGEBRA LECTURES Coprights and Author: Kevin Pinegar Chapter 0 PRE-ALGEBRA TOPICS 0.4 Polnomials and Factoring Polnomials And Monomials A monomial is a number, variable or a product of numbers and variables: 7, x, and x are all monomials A polnomial is a variable expression in which the terms are monomials. 7, x, x-5, and x -x +4x-1 are all polnomials A monomial is a polnomial of one term. Examples: x, 7, -5x Consider the Monomial: -5x The Coefficient is -5, The variable (or base) is x, and the exponent:= A binomial is a polnomial of two terms. Examples: x+7, x -5, 4a- Consider the Binomial: x -5: The first term is x and the second term is -5 A trinomial is a polnomial of three terms. Examples: x -x+4, x +x-1, x++4z Consider the Trinomial: x -x+4 First Term is x Second Term is -x Third Term is 4 If a polnomial has more than terms we simpl call it a general polnomial. Example: 5x 4 +x -x +4x-1 The Degree of a Monomial The degree of a monomial is the sum of its exponents. Identif the degree of the monomials below. 4x Degree is. 5x Degree is +=4. xz Degree is 1+1+1= x 0 Degree is 0. The Degree of a Polnomial The degree of a polnomial is same as the highest degree monomial. Identif the degree of the polnomials below. 4x x 7 x Degree is. See nd term. 4x x 7x Degree is 5. See last term. x 5 Degree is 1. See 1 st term. Polnomials are generall written in descending order (highest degree to smallest degree). Below are polnomials in descending order. 5x 4 +x -x +4x-1 x -x+4 x+7 Note: After functions are discussed, we can talk about polnomial functions. An example of a polnomial function is P(x)= x -x+4 1

2 Operations On Polnomials Adding and Subtracting Polnomials In order to add or subtract polnomials, we need to know what terms are like terms and what terms are unlike terms. The ke is to compare the variable portion. If the variable portion is exactl the same, the terms are like terms. Otherwise the terms are unlike. Like terms can be added or subtracted whereas unlike terms cannot. Each set of terms below are like terms: {x and x}, {5, and - }, {x, x and 0.5x } Each set of terms below are unlike terms: {x and 5}, {5 and - }, {x and x } Add or Subtract the Following (-)+(-4) =++(-)+(-4) = 5-7 (5x-7)-(x-8) =5x-7-x+8 =5x-x-7+8 =x+1 (4c +c-)+(c -c+) = 4c + c +c-c-+ = c +c-1 (-z -4z+7)+(z +z-1) (z -z+7) = -z -4z+7+z +z-1 z +z-7) = -z -z -4z+z+z = -5z +z-1 (b +b-5)-(b -4b-9) = b +b-5-b +4b+9 = b -b +b+4b-5+9 =7b+4 Multipling Polnomials Before we multipl, let s make sure we know the base and exponent. Term Base Exponent x 5 x 5 (-) 4-4 (5) Monomial Multiplication (using the distributive propert) 5x (x -x+) (4a -a-)(7a) = 5x (x ) 5x (-x) 5x () =4a (7a)-a(7a)-(7a) = 15x +10x 5x =8a 14a -1a General Polnomial Multiplication (using multiple distributive properties) (x+)(x -4x+7 ) (x +4x-6)(x-) = x(x )+x(-4x)+x(7 )+(x )+(-4x)+(7 ) Let s do this in our head. = x -4x +7x +6x -8x +14 =6x -9x +8x -1x-1x+18 =x +x -x +14 = 6x -x -4x+18

3 Binomial Multiplication {FOIL} When multipling two binomials together we can use FOIL method. F=product of first terms, O=product of outer terms, I=product of inner terms, L=product of last terms. EXAMPLE: (x-)(x+5) FIRST Product OUTER Product INNER Product LAST Product x(x)=6x x(5)=15x -(x)=-4x -(5)=-10 Result: 6x + 15x 4x 10 Combine Like Terms: 6x + 11x 10 Multipl The Binomials (7x+)(x-5) = 14x -5x+6x-15 = 14x -9x -15 (a b-c)(5a-bc) = 15a b a b c 10ac + bc (no like terms) Special Products: There are special products to use as a shortcut when multipling two binomials. The first leads to difference of squares and the other two are for squaring a binomial. Difference Of Squares: (A+B)(A-B)=A B Examples: (x+7)(x-7) =(x) (7) = 9x 49 (x-5 )(x+5 ) =(x) (5 ) = 4x 5 4 (5ab+6)(5ab-6) = (5ab) (6) =5a b 6 Squaring A Sum: (A + B) = A + AB + B Examples: (x+) or (x+)(x+) = (x) +(x)() +() = 9x + 1x + 4 (5x +) or (5x +)(5x +) = (5x ) + (5x )() + = 5x 4 +10x + Squaring A Difference: (A B) = A AB + B (These last two are kind of the same formula except for the middle term) Examples: (x-7) or (x-7)(x-7) = (x) (x)(7) + (7) = 4x 8x +49 (4m -5n )(4m -5n ) = (4m ) (4m )(5n ) + (5n ) = 16m 4 40m n + 5n 6 Factoring Thought: Later in this section we will learn to factor trinomials. Just to give ou a heads up, let me show ou the factors of some examples that we alread multiplied. We will just work backwards. Since (x-)(x+5) = 6x +11x-10, then 6x +11x-10, factors as (x-)(x+5) Since (x+7)(x-7) = 9x 49, then 9x 49 factors as (x+7)(x-7) Since (x-) = 9x -1x+4, then 9x -1x+4 factors as (x-) Applications: 1. Find the area of a square whose sides have length of (x-7) inches. A=s. Find the volume of a box, with dimensions; x, 5x-1, x+. V=LWH. Find the perimeter of a triangle with sides: x, x+5, x-4. P=a+b+c Solution: (x-7) = 4x -8x+49 Solution: x(5x-1)(x+) =x(10x +1x-) =0x +9x -9x x+(x+5)+(x-4) =6x+1

4 Factoring Polnomials The Greatest Common Factor {GCF} Before we can discuss factoring, we must know how to find a GCF. The GCF for real numbers is the largest number that divides evenl into all terms. For a variable, it is the largest power of the variable that divides evenl into all terms. What is the GCF: {4,16,} Answer: 8 What is the GCF: {0,40,50} Answer: 10 What is the GCF: {1,-9,-6} Answer: What is the GCF: {x,x 5, x } Answer: x What is the GCF: {x,x, x } Answer: x What is the GCF: {(x+)(x-1), (x+)(x-1) } Answer: (x+)(x-1) Putting It All Together: What is the GCF: {4x,16x 5, x } Answer: 8x What is the GCF: {0 x,40 x,50 x } Answer: 10x What is the GCF: {1(x+)(x-1),-9(x+)(x-1) } Answer: (x+)(x-1) Factoring Out the Greatest Common Factor {GCF} When the GCF is factored from a polnomial, the resulting factor is determined b dividing the GCF into each term. On the first couple of examples, I will show the steps, but ou can probabl calculate the quotients in our head. Factor out the GCF: Solution: GCF= 8x 5 4x 16x x Factor out the GCF: 0x 40x 50x 4x 16x 5 x 8x x 4x 8x x 4x 8x Solution: GCF= 10x 0x 40x 4x 8x 50x 16x 8x 5 x 8x 0x 10x 10x 10x x 4 5x Factor out the GCF: Solution: GCF= ( 4 5 ) ( 5 4) Factoring Out the NEGATIVE Greatest Common Factor {GCF} Sometimes we want to factor out the negative GCF. If the first term is negative, we might do this so the first term in the factor is positive. Factor out the Negative GCF: Solution: Negative GCF= -x 6x 9x x 6x 9x x x(x x 1) Factor out the GCF: Solution: Negative GCF= -4x 8x 0x 16x 8x 0x 50x 4x(x 5 4) 40x 10x 50x 10x Sometimes the GCF involves another polnomial Factor out the GCF: 1( x )( x 1) 9( x )( x 1) Solution: GCF= 1( x )( x 1) ( x )( x 1) ( x )( x 1) 9( x )( x 1) ( x )( x 1) (4 x ) ( x )( x 1) ( x 7) 4 ( x 1) 4

5 Factoring b Grouping The above example was an example of factoring b grouping. With this method, ou have two or more terms and each term contains a common polnomial factor that we can factor. Example: a(x-) + b(x-) a(x-) + b(x-) {The green expression is 1st term and blue is nd term.} Note that each term has a common factor of (x-). We can factor (x-) from each term. a(x-) + b(x-) = (x-) (a+b) = (x-) (a+b) Factor Each Expression That Is Alread Grouped 4x(x+5) (x+5) x(+1) + a(+1) GCF = (x-5) GCF= (+1) 4x(x+5) (x+5) =(x-5)(4x-) a(a-b) 5b(a-b) GCF= (a-b) x(+1) + a(+1) =(+1)(x+a) x (x-) + (x-) GCF= (x-) a(a-b) 5b(a-b) x (x-) + (x-) =(a-b)(a-5b) =(x-)(x +1) In the above examples, the terms were alread grouped for ou. Most of these tpes of problems do not begin this wa. You actuall have to group two pairs of terms and factor from each pair to see the common factor. Example: 9p-9q+mp-mq = 9(p-q) + m(p-q) = (p-q)(9+m) Factor B Grouping 6x + x 10x 5 =x(x+1) 5(x+1) =(x+1)(x-5) x+xz 5 5z =x(+z) 5(+z) =(+z)(x-5) 5+15 = (-) 5(-) =(-)( -5) Factoring Trinomials There are two cases for factoring trinomials in the form ax +bx+c 1. Eas case, where a=1. Example: x +1x-100. Harder case, where a1. Example: 5x +6x+1 Case 1: Factoring Trinomials of the Form: x +bx+c The factors in this case will be (x+m)(x+n) where mn=c and m+n=b Example: x +1x-100. Find two numbers whose product is -100 and sum is 1. If ou search the factors of -100 long enough, ou will come up with m=5 and n=-4. So x +1x-100=(x+5)(x-4) {You can check it using the foil method} Factor Each Trinomial x -15x+56 (-8)(-7)=56 and (-8)+(-7)=-15 x -15x+56= (x-8)(x-7) x 49 (here b=0) (-7)(7)=-49 and (-7)+7=0 x 49 = (x+7)(x-7) a +5a+100 (0)(5)=100 and 0+5=5 a +5a+100= (a+0)(a+5) (4)=16 and 4+4= = (+4)(+4) x -10x-96-16(6)=-96 and (-16)+(6)=-10 x -10x-96 = (x-16)(x+6) Since we have in last term, the factors must include. x 18x + 81 (-9)(-9)=81 and (-9)+(-9)=-18 x 18x + 81 = (x-9)(x-9) 5

6 Case : Factoring Trinomials of the Form: ax +bx+c There are two techniques for these tpes. 1. Trial and Error: The trial and error method is exactl what is sas. You tr a combination of factors of the first term and factors of the last term until ou find the factors that also give ou the correct middle term.. Grouping Method: I will discuss this method at the end. It requires a little more structure and the proper steps. Choose the method that works best for ou. Factor Each Trinomial Using Trial And Error Trinomial Trial Factors Check the Middle Term 5x +6x+1 Factors of 5x = {5x,x} Factors of 1 = {-1,-1}, {1,1} p -16p+5 Factors of p = {p,p} Factors of 5 = {1,5}, {-1,-5} 10t -11t+ Factors of 10t ={10t,t}, {5t,t} Factors of = {1,}, {-1,-} 15x -9x+1 Factors of 15x ={15x,x}, {5x,x} Factors of = {1,1}, {,6},{,4} Trials (5x-1)(x-1) (5x+1)(x+1) Trials (p+5)(p+1) (p-5)(p-1) (p+1)(p+5) (p-1)(p-5) Trials (10t+1)(t+) (5t+1)(t+) (5t+)(t+1) (5t-)(t-1) Trials (15x-)(x-1) (5x-)(x-1) (5x-)(x-6) (5x+)(x+4) (5x-)(x-4) Middle Term = 6x -5x-1x=-6x (NO) 5x+1x=6x (YES) Answer: 5x +6x+1=(5x+1)(x+1) Middle Term = -16p p+5p=8p (NO) -p-5p=-8p (NO) 15p+1p=16p (No, but close) -15p-1p=-16p (YES) Answer: p -16p+5 = (p-1)(p-5) Middle Term = -11t 0t+1t=1t (NO) 15t+t=17t (NO) 5t+6t=11t (NO)** -5t+(-6t)=-11t (YES) Answer: 10t -11t+= (5t-)(t-1) **when middle term is opposite, change the signs of both nd factors Middle Term = -9x -180x+(-x)=-181x (Heck No) -60x+(-15x)=-75x (Nope) -0x+(-6x)=-6x (NO) 0x+9x=9x (NO) -0x+(-9x)=-9x (YES) Answer: 15x -9x+1 = (5x-)(x-4) NOTE: Not all trinomials will factor. If we cannot find a set of factors, we sa the trinomial is not factorable over the set of real numbers. Bummer, huh? Factoring Trinomials Using Grouping Method: For ax +bx+c, find two numbers m and n whose product is ac and sum is b. Then replace the middle term using m and n as coefficients. After ou do that, factor b grouping. Example: p -16p+5 ; ac=15 and b=-16. If we use m=-15 and n=-1, then mn=-15 and m+n=-16. Replace -16p with -15p-1p then use grouping method. Solution: p -16p+5= p -15p-1p+5 = p(p-5)-1(p-5)= (p-5)(p-1) Example: 10t -11t+ ; ac=0 and b=-11. If we use m=-6 and n=-5, then mn=0 and m+n=-11. Solution: 10t -11t+ = 10t -6t-5t+= t(5t-)-1(5t-) = (5t-)(t-1) Example: 10x -17x-0 ; ac=-00 and b=-17. If we use m=8 and n=-5, then mn=-00 and m+n=-17. Solution: 10x -17x-0 = 10x +8x-5x-0 = x(5x+4)-5(5x+4) = (5x+4)(x-5) The Grouping Method is more difficult if mn is large, but it alwas works if properl done. 6

7 Special Factoring Formulas EXAMPLES Difference of Perfect Squares: (A B ) = (A+B)(A-B) (5x -16)=(5x+4)(5x-4) Sum of Perfect Squares do not factor. (5x +16) not factorable Difference of Perfect Cubes: (A B ) = (A-B)(A + AB + B ) (8x -7)=(x-)(4x +6x+9) Sum of Perfect Cubes: (A + B ) = (A+B)(A AB + B ) (8x +7) )=(x+)(4x -6x+9) Perfect Square Trinomials: A + AB + B = (A B) A - AB + B = (A + B) (5x +40x+16)=(5x+4) (5x -40x+16)=(5x-4) Factor Each Using An Appropriate Formula Difference of Squares 1) x -6 = x 6 =(x+6)(x-6) Perfect Square ) x -16x+64=x -(8x)+8 =(x-8) Diff of Cubes ) -64= 4 =(-4)( +4+16) Sum of Cubes 4) 8p +7q = (p) + (q) (p+q)(4p -6pq+9q ) Perfect Square 5) 49x +8x+4 =(7x) +(7x)()+() =(7x+) Sum of Squares 6) x +4 not factorable Factoring COMPLETELY: When factoring, ou should alwas factor completel, but when ou see the words factor completel an alarm should go off that ou ma have to use multiple factoring techniques. This could mean ou have to factor out the common factor first, then factor a trinomial or it could mean that the factors can be factored again. Example: = 4( +-18) = 4(+9)(-6) Example: x 4 16 = (x +4)(x -4) = (x +4)(x+)(x-) Factoring COMPLETELY 1) m 4-56 =(m +4)(m -4) =(m +4)(m+)(m-) ) a +1ab-54b = (a +7ab-18b ) =(a+9b)(a-b) ) x +x-6 = (x +x-1) =(x+4)(x-) 4) 7-x =(6-x ) =(6+x)(6-x) 7) ab +7ab-8a = a(b +7b-8) =a(b-8)(b+1) 5) 4(x-5)-x (x-5) =(x-5)(4-x =(x-5)(+x)(-x) 8) 1-4x-x =-1(x +4x-1) = - (x+6)(x-) 6) 70-80x =10(7-8x ) =10(-x)(9+6x+4x ) 9) x 6-6 =(x + )( x - ) =(x+)(x +x+ ) (x-)(x -x+ ) Solving Equations b Factoring: We will just introduce the concept here. This is covered in more detail later. We will use the Zero Product Propert given below: IF AB=0, the either A=0 or B=0. Example: (x+)(x-5)=0, then either (x+)=0 or (x-5)=0. Solving ields x=- or x=5. 1) x -6=0 (x+6)(x-6)=0 x+6=0 or x-6=0 x=-6, x=6 4) p -16p+5=0 (p-1)(p-5)=0 p-1=0 or p-5=0 p= 1/, p=5 ) 4(x-5)-x (x-5)=0 (x-5)(+x)(-x)=0 x-5=0, +x=0, -x=0 x=5, x=-, x= 5) x =9x x 9x=0 x(x -9)=0 x(x+)(x-)=0 x=0, x+=0, x-=0 x=0, -, ) 15x -0x=0 5x(x-4)=0 5x=0 or x-4=0 x=0, x= 4/ 6) 9-1+4=0 (-)(-)=0 -=0 = =/ 7

8 Applications: Set up an equation and solve for the answer. a) The square of a negative number is fifteen more Solution: Let x= the number. than twice the number. Find the number. x =x+15 x -x-15=0 (x-5)(x+)=0 x-5=0, x+=0 x=5, x=- b) The sum of numbers is 8. The sum of the squares of the numbers is 4. Find the numbers. c) The width of a rectangle is 5 ft less than the length. The area is 176 ft. Find the length Factoring Guidelines Summar I. Factor Out The GCF: x 1x 6x x( x 4x ) Answer: x=- Solution: x= one number and 8-x is other. x +(8-x) =4 x x+x =4 x -16x+64=4 x -16x+0=0 (x -8x+15)=0 (x-5)(x-)=0 x-5=0, x-=0 x=5, x= x=5, 8-x= Solution: L=length, L-5=width {A=LW} L(L-5)=176 L -5L=176 L -5L-176=0 (L-16)(L+11)=0 L-16=0, L+11=0 L=16ft is onl viable answer. W=16-5=11ft II. Four Terms: Factor B Grouping: x x 4x 6 x(x ) (x ) ( x )(x ) III. Two Terms: 1) Difference of Squares? a -b =(a+b)(a-b) 49x 16 (7x 4)(7x 4) ) Difference of Cubes? a -b = (a-b)(a +ab+b ) 7x 64 (x 4)(9x 1x 16 ) IV. Terms 1) Perfect Square Trinomial? a +ab+b =(a+b) 4x 8x 49 (x 7) ) Trial and Error 6x x 1 (x 4)(x ) ) Sum of Squares Does Not Factor 49x 16 49x 16 5) Does The Shortcut Appl? (Case 1): x 7x 10 ( x 5)( x ) Factor Completel; Can The Factors Be Further Factored? 4 ( x 56) ( x 16)( x 16) x 6x 45x 4) Sum of Cubes? a +b = (a+b)(a -ab+b ) 7x 64 (x 4)(9x 1x 16 ) ) Perfect Square Trinomial? a -ab-b =(a+b) 5x 0x 9 (5x ) 4) OR Grouping 6x x 1 6x 9x 8x 1 x(x ) 4(x ) (x 4)(x ) ( x 16)( x 4)( x 4) x( x x 15) x( x 5)( x ) Note: The formulas for cubing a binomial are: (A+B) =A +A B+AB +B and (A-B) =A -A B+AB -B 8