POD Combine these like terms: 1) 3x 2 4x + 5x 2 6 + 9x 7x 2 + 2 2) 7y 2 + 2y 3 + 2 4y + 5y 2 3) 5x 4 + 2x 5 5 10x 7x 4 + 3x 5 12 + 2x 1
Definitions! Monomial: a single term ex: 4x Binomial: two terms separated with either addition or subtraction. ex: 3xy + 5t Trinomial: three terms separated with either addition or subtraction. ex: 5y 10y 2 + 6x Polynomial: a generic word for the sum of monomials (monomial, binomial, trinomial and beyond are all polynomials). 2
Degree of a monomial: Definitions! Degree of a polynomial: ex: 4y 5xz 6.5 7a 3 + 9b 6x 2 + 4x + x + 3 3
Standard Form and Leading Coefficients Standard form: writing polynomials in order from greatest degreed term to least degreed term. ex: bad ex: good! 2x 5x 2 + 7 + 4x 3 4x 3 5x 2 + 2x + 7 Leading Coefficients: the first coefficient in standard form ex: 4x 3 5x 2 + 2x + 7, leading coefficient: 4 Put into standard form: a) 8 2x 2 + 4x 4 3x b) y + 5y 3 2y 2 7y 6 + 10 4
Adding and Subtracting Polynomials If you're adding polynomials: just add the like terms! (2x 2 + 5x 7) + (3 4x 2 + 6x) If you're subtracting polynomials: distribute the negative then add! (3 2x + 2x 2 ) (4x 5 + 3x 2 ) ex: (3y + y 3 5) + (4y 2 4y + 2y 3 + 8) FINAL ANSWERS ARE IN STANDARD FORM! (4x 3 3x 2 + 6x 4) ( 2x 3 + x 2 2) 5
Practice 1) (7y 2 + 2y 3) + (2 4y + 5y 2 ) 2) (6y 2 + 8y 4 5y) (9y 4 7y + 2y 2 ) homework: 8.1, page 468, 1 18, 34 37 6
1) yes, 3, trinomial 2) yes, 2, trinomial 3) yes, 2 monomial 4) no 5) yes, 5, binomial 6) no 7) Homework Answers 7
Adding Questions to Tests and Quizzes 8
1) 3x(2xy) POD Multiply 2) 4(3x 2 + 2x 1) 3) 2x 3 (10x 2 y 3 ) 9
Multiplying a polynomial by a monomial This combines the distributive property and the exponent properties from chapter 7 3x 2 (7x 2 x + 4) ex: 5a 2 ( 4a 2 + 2a 7) 6d 2 (3d 4 2d 3 d + 9) 10
Multiplying and Simplifying 3(5x 2 + 2x + 9) + x(2x 3) 4d(5d 2 12) + 7(d + 5) 11
Solving Crazy Long Problems Simplify the left and right sides. Solve! 2a(5a 2) + 3a(2a + 6) + 8 = a(4a + 1) + 2a(6a 4) + 50 7(t 2 + 5t 9) + t = t(7t 2) + 13 12
Homework Answers 13
POD 1) x(x 4) + 5(x 4) 2) 3x(2x 5) 4(2x 5) 14
(x 2)(x + 5) Multiplying Binomials (2x 4)(5x + 3) 15
Multiplying Binomials (x 2)(x + 5) F O I L (2x 4)(5x + 3) 16
Multiplying Binomials (x + 4)(2x 7) F O I L ( 3x + 2)(x 9) (2x y)(3x + 4y) 17
1) (2y 7)(3y + 5) Practice 2) (4a 5)(2a 9) 3) (4x 5z)(3x 4z) 18
Homework Answers 19
1) (3x 4)(2x + 5) POD 2) 4x(x 2 + 2x 6) + 7(x 2 + 2x 6) 20
Binomial and Trinomials (6x + 5)(2x 2 3x 5) (3x 5)(2x 2 + 7x 8) 21
Trinomial and Trinomials (2y 2 + 3y 1)(3y 2 5y + 2) (x 2 + 2x 3)(4x 2 7x + 5) 22
` (x + 4)(3x 2 3x + 8) (2x 2 + 5x 1)(6x 2 + 3x + 2) 23
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1) (x + 3)(x + 3) POD Use FOIL 2) (4x 5)(4x 5) 3) (3x + 7)(3x 7) 25
(a + b) 2 Square of a Sum (x + 3) 2 (2x + 6) 2 26
(a b) 2 Square of a Difference (y 8) 2 (5x 3) 2 27
Product of a Sum and Difference (a + b)(a b) (x + 7)(x 7) (5x + 8y)(5x 8y) 28
Practice (3x + 1) 2 (4x 3) 2 (9x+2)(9x 2) 29
Homework Answers 30
POD 1) 3xy( 2x 2 + 4x 2 y 5xy 3 ) 2) Solve for d: 2d 12 = 0 3) Solve for x: 5x + 65 = 0 31
At the end of notes, you will be able to: Factor an algebraic expression Simplify a polynomial by grouping Solve for zero ed solutions 32
Factoring Algebraic Expressions To "factor" a problem means to find the LARGEST common factor (GCF) in an addition/subtraction problem. I like to refer to it as "Reverse distributive property" 32x 4 y 3 48x 2 y 3 + 16y 2 33
Examples 3x 2 + 15xy 9x 3 y 6xy 2 34
Factoring using grouping We use grouping when we have four terms. How it works: 4qr + 8r + 3q + 6 group the first two and last two terms 4qr + 8r + 3q + 6 factor out the GCFs of both sets 4r(q + 2) + 3(q + 2) if done correctly, what's left over are common factors (q + 2)(4r + 3) factor out (q + 2) and you have the final answer 35
Grouping Examples 3mp + 15p 4m 20 2xy + 7x 2y 7 36
Grouping with Additive Inverses 2mk 12m + 42 7k c 2cd + 8d 4 37
(3x 8)(2x + 7) POD (2x 5)(x 2 8x + 9) (3x 5y)(3x + 5y) 38
Solving for zero'ed equations When solving, set factors = to 0 (2x + 6)(3x 15) = 0 c 2 = 3c 3n(n + 2) = 0 39
Homework Answers 40
1) (x + a)(x + b) POD 2) (x + c)(x d) 3) (x + 3)(x + 4) 41
As you learned in the POD: Reverse FOIL The middle term (typically 'x') is the SUM of the second terms. The last term (typically a constant) is the PRODUCT of the second terms. (x + 3)(x + 4) = x 2 + 7x + 12 x 2 + (3 + 4)x + (3*4) (x 7)(x + 8) = x 2 + 1x 56 x 2 + (8 + 7)x + (8* 7) 42
Reverse FOIL We are going to look at the middle and last terms to help us figure out how to "reverse foil" x + 11x + 24 What two numbers multiply to be 24... that add up to be 11? (x )(x ) 43
Reverse FOIL We are going to look at the middle and last terms to help us figure out how to "reverse foil" x 2 + 3x 18 What two numbers multiply to be 18... that add up to be 3? (x )(x ) 44
Reverse FOIL We are going to look at the middle and last terms to help us figure out how to "reverse foil" x 2 12x + 27 What two numbers multiply to be 27... that add up to be 12? (x )(x ) 45
x 2 + bx + c Patterns! (x + y)(x + z) x 2 + bx c (x y)(x + z) when... x 2 bx c (x y)(x + z) when... x 2 bx + c (x y)(x z) 46
Practice x 2 3x + 2 y 2 + y 42 m 2 + 3m 54 x 2 4x 5 y 2 + 13y + 40 36 + 5p + p 2 47
Homework Answers out of 12 48
POD 1) Solve for x: (x 5)(x + 9) = 0 2) Factor: x 2 7x 30 3) Use grouping to factor: 2x 2 + 4x 5x 10 49
Solving Quadratic Equations When given in the form ax 2 + bx + c = 0, Factor the problem then solve (like in 8.5) x 2 + 6x = 27 x 2 3x = 70 x 2 + 3x 18 = 0 50
Beginning to Factor Quadratics 2x 2 + 5x + 3 When given a quadratic with a Leading Coefficient bigger than 1, we have to change our system (only a little) 2x 2 + 5x + 3 1) Multiply the Leading Coefficient and the constant 2) Just like yesterday, thing what to factors will add up to equal the middle term. 2 * 3 = 6, 2 + 3 = 5 3) Split up the middle term (5x) into the addition problem (2x + 3x) that you just discovered. 2x 2 + 5x + 3 = 2x 2 + 2x + 3x + 3 4) Use grouping to finish. 51
7x 2 + 29x + 4 Extra Practice 5x 2 + 13x + 6 6x 2 + 22x 8 3x 2 17x + 20 52
Homework Answers out of 14 53
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1) (3x 4)(3x + 4) POD 2) ( 5y + 2)( 5y 2) 3) (9z 7w)(9z + 7w) 55
Difference of Squares For any problem: (a b)(a + b) = (a 2 b 2 ) AND for in reverse: (a 2 b 2 ) = (a b)(a + b) This is called the DIFFERENCE of SQUARES If two perfect squares are being subtracted, they can be split into their SQUARE ROOT'S conjugates ex: (16a 2 9) (x 2 81) (36y 4 144b 2 ) 56
DOUBLE Difference of Squares A DOUBLE difference of squares can occur when one of your conjugates is ALSO a difference of squares: (16a 4 1) (625 b 4 ) 57
Factorable Problems Sometimes you don't see a difference of squares until you factor out a GCF: (5x 5 45x) 7x 3 + 21x 2 7x 21 2a 4 50 2m 3 + m 2 50m 25 58
Homework 8.8: page, 518: 2 12 even, 15 36 every 3 (15, 18, 21...) This is 14 problems 59
Homework Answers out of 14 60
1) (2x + 3) 2 POD 2) (x 8) 2 3) (4x 2 + 5) 2 61
Perfect Square Trinomials Recall that perfect squares are: The first term squared, the doubled product of the first and second term and the second term squared (2x + 3) 2 = (2x) 2 + 2(3)(2x) + (3) 2 (x 8) 2 = (x) 2 + 2(8)( x) + (8) 2 (4x 2 + 5) 2 = (4x) 2 + 2(5)(4x) + (5) 2 62
Perfect Square Trinomials To work backwards, we need to CHECK IF: The square root of the first term and the square root of the last term is the doubled product of those two answers Are these Perfect Square Trinomials? If so, factor them! 4y 2 + 12y + 9 9x 2 6x + 4 9y 2 + 24y + 16 63
Perfect Square Trinomials Determine whether these are perfect squares. If so, factor them! 2y 2 + 10y + 25 25x 2 30x + 9 49x 2 + 42x + 36 x 2 24x + 144 64
Different Factoring Methods 65
5x 2 80 Factor to the best of your ability 9x 2 6x 35 2x 2 32 12x 2 + 5x 25 66
Homework Answers out of 12 67
Solve for x: 1) 3x 4 = 0 POD 2) 2x + 5 = 0 3) Factor: 9x 2 48x + 64 68
9x 2 48x = 64 Solving For X x 2 + 12x + 36 = 0 (y 6) 2 = 81 (x + 6) 2 = 12 69
Homework Answers out of 10 70
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