Polynomials Lesson 5.0 Re-Introduction to Polynomials Let’s start with some definition. Monomial - an algebraic expression with ONE term. --------------------------------------------------------------------------------------------- Binomial - an algebraic expression with TWO terms. --------------------------------------------------------------------------------------------- Trinomial - an algebraic expression with THREE terms. --------------------------------------------------------------------------------------------- Polynomial is a general description on any algebraic expression with 1 term or more. To add or subtract polynomials, we combine like terms. Like terms - same variables raised to the same exponents --------------------------------------------------------------------------------------------- The term with the greatest exponent sum determines the degree of polynomial for an algebraic expression. Eg1. Identify the type and the degree of each polynomial. a) 2x + 3 b) -5a2b – 2 + a5 c) 2p2 + 3p3q2 – p2 d) -4yx2 + 3 – 2x2y + 5x2y2

Eg2. Simplify. a) (3x2 – 6x + 7) – (7x2 – 2x + 9) b) (a2b2 – b2) + (3b2a2 + 2a2) c) -8(2k – 3) + 11(k – 2) d) 3a2(-a3 + b – 4a2b) – 2b(a2 – 5) To simplify expressions with a multiplication of two monomials, we use BEDMAS . Eg1. Simplify.

a) )5)(3( 2xx − b) )3)(6( 32 pqqp − c) 283

3254

)10()2)(5(

nmmnnm

−

−

Practices: Worksheet # 1cf, 2gh, 3gh, 4bf, 5df, 9ef, 10ce, 11ef, 18bf, 21ad, 25ad, 27bf, 31be

Polynomials Lesson 5.1 Multiplying Polynomials To multiply a monomial by a polynomial, we use . Before we continue, let’s look at a square with unknown side. Eg1. A square has four equal sides, with length x cm. a) Find its area. using algebra tiles: using distributive property: b) If the square is to be stretched sideway by 1 cm. Find its new area. using algebra tiles: using distributive property: c) Further, the height of the rectangle is lengthened by 2 cm. Find its final area. using algebra tiles: using distributive property:

Eg2. Use algebra tiles to determine the product (x + 3) (x – 1). Eventually, multiplying polynomials would be more efficient with the distributive property. Eg3. Expand )2( +xx .

Using similar idea, we can distributive a binomial into another binomial. Eg4. Expand ( )( )32 +− xx .

The expansion above is so frequently used that there is an acronym for it. FOIL: ____________________________________________________________________ Eg5. Expand and simplify. a) (x – 3)(2x + 5) b) (4a – 3b)2

To multiply a binomial by a polynomial, we can extend the idea of FOIL. Eg6. Expand. a) (3x2 – 5) (5x2 – 2xy + y2)

b) (4x – 1) (3x2 + 2x – 5) c) (2x + 1)3 Eg7. A cube with unknown side (x cm) is to extend its length by 3 cm and to double its

height. Express its volume in terms of x if the width is to shorten by 2 cm. Practices: MHR p.209 # 1adf, 2, 3df, 4, 5bd, 6ade, 7, 10, 11, 13, 15, 17, 19 Polynomials

Lesson 5.2 Common Factor Factors that are shared by all terms in an algebraic expression are said to be common factor. That is, a common factor must be divisible by all terms in the expression. Eg1. Consider the expression 18x – 12y. a) State the common factor for 18x and 12y. b) Factor the expression 18x – 12y. To factor an algebraic expression completely, we can first identify the greatest common factor (GCF) of all the terms in the expression. Once we factorize the expression with the GCF, the expression would be easier to deal with. Eg2. A rectangular prism has length l , width w, and height h. Express its surface area in two

different ways.

Eg3. Factor the following completely. a) 2x – 8 b) 24x3y5 + 18x2y c) 55JΔk2 – 10kΔJ

Idea: Factor out GCF for all coefficients, and use lowest exponent for each variable. Eg4. Factor completely. a) 24x2(x – 4) – 42xy(x – 4) b) a3b2(2c – d) + a2b3(d – 2c) Eg5. Factor by first grouping similiar terms together. a) 2 8 16 2y xy x y+ + + b) 2 8 7 28cx x c− + − c) 2 8 2 16y xy y x+ + + Eg6. Joey the Little Monster went trick or treat on the Halloween night. He got 12 Snickers,

24 Starburst, 28 Mars, and 36 Kit-Kat. His plan is to divide them into equal piles with the same number of each type, and re-gift the rest to the food bank.

a) In how many piles can he divide his candies into? b) How many of each type is he giving to the food bank? Practices MHR p.220 # 2ce, 4be, 5, 6, 7abc Worksheet #1(all), 2(all)

MHR p.220 # 10, 12, 13, 15, 16, 18, 19ab Worksheet 1. Factor the following. a) ( ) ( )3 2 4 2y y y− + − b) ( ) ( )5 4 2 4a a a− + −

c) ( ) ( )2 5 2m x y y x− − − d) ( ) ( )7 10w x w w x+ − +

e) ( ) ( )24 4 5x x x+ − + f) ( ) ( )22 5 5 2m n a n m bc− − −

g) ( ) ( ) ( )2 3 7m a b n a b a b− − − − − h) ( ) ( ) ( )22 3 2 5 3 2 9 2 3x a b x a b b a− + − − − 2. Factor the following by grouping. a) 2 3 3x x xy y+ + + b) 5 10 2am a bm b+ + + c) 3 2 1x x x+ + + d) 23 6 5 10x xy x y− + − e) 25 10 3 6m mn m n+ − − f) 22 6 3 9a ab a b− − + g) 6 4 22 10 5y y y+ − − h) 5 3 2 1x x x+ − − Answers 1. a) ( )( )3 4 2y y+ − b) ( )( )5 2 4a a− −

c) ( )( )5 2m x y+ − d) ( )( )7 10w w x− +

e) ( )( )24 5x x+ − f) ( )( )22 5m n a bc− +

g) ( )( )2 3 7m n a b− − − h) ( )( )22 5 9 3 2x x a b+ + −

2. a) ( )( )3x x y+ + b) ( )( )2 5 1a b m+ +

c) ( )( )2 1 1x x+ + d) ( )( )3 5 2x x y+ −

e) ( )( )5 3 2m m n− + f) ( )( )2 3 3a a b− −

g) ( )( )4 25 2 1y y− + h) ( )( )3 21 1x x− +

Polynomials Lesson 3a Factoring Trinomials Eg1. Expand by FOIL. a) (x + 3)(x + 5) b) (k – 2)(k + 7) In general, expanding (x + a) (x + b) results in x2 + __________ x + ___________ Expanding a pair of binomials often results in a trinomial (not always as we will see next class). To factor a trinomial as a product of two binomials, we reverse the steps. 1. Find two numbers whose product equals the last term; 2. Check if the sum/difference of these two numbers would equal the middle term; 3. Use these two numbers to factor the trinomial. Eg2. Factor the given trinomials. a) x2 + 11x + 24 b) a2 + 5a – 6 c) 10y2 – 20y – 150 d) 22 26 60ab ab a− −

Remember: Always check for common factor before factorizing the trinomial. Eg3. Factor the following polynomials. a) 3k4 + 3k2 – 36 b) 5a2b2 – 65ab – 150 c) 14m2 + 5mn – n2 d) (x + y)2 – 8(x + y) + 15 Eg4. For which integral values of k can the trinomial be factored? x2 + kx – 6 Practices Worksheet # (3, 5, 7)behk, 13ac, 14af, 18, 19abef, 20cd, 21cdf MHR p.235 # 8a, 1(all)

Answers:

Wor

ksh

eet

Polynomials Lesson 3b Factoring Trinomials with Leading Coefficient Eg1. Factor the following. a) 1 – 5c + 6c2 b) x2 + 2xy – 3y2 / \ / \ –2c –3c -y +3y = (1 – 2c) (1 – 3c) = (x – y)(x + 3y) c) a2b2 + 5ab – 6 d) 2k3 + 18k2 + 36k / \ +6 -1 Factor out the 2k first! = ( ? + 6 ) ( ? – 1 ) = 2k (k2 + 9k + 18) / \ = (ab + 6) (ab – 1) +3 +6 = 2k (k + 3) (k + 6) When we are to factor trinomials with a leading coefficient: ax2 + bx + c We use the method of decomposition : It begins by multiplying the leading coefficient (a) to the last term (c).

Eg2. Factor a) 2x2 + 7x + 6 b) 3x2 – 10x + 8

12 use the same method as in (a) / \ ans: (3x + 2) (x – 4) +3 +4 = 2x2 + 3x + 4x + 6 (emphasize we only decompose the middle term (7x) into two pieces) as we don’t change the front (2x2) or the back (6) = 2x2 + 3x + 4x + 6 (split the 4 terms into 2 groups – then factor by grouping) draw a line to indicate the two groups = x (2x + 3) + 2 (2x + 3) (identify the common factor for the first 2 terms & the last 2 terms) = (2x + 3) (x + 2) (factor out 2x + 3, and copy the rest to form the second bracket) Idea: Need two terms whose product = a x c whose sum = b Eg3. Factor completely.

a) 4x2 + 4x – 3 b) 2k2 – 14k + 12 You can get students to try these now. Remind them to identify the common factor if there are any! = (2x + 3) (2x – 1) = 2 (k2 – 7k + 6) = 2 (k – 1)(k – 6) c) 10xy2 – 22xy + 4x d) 8c2 + 18cd – 5d2 = 2x (5y2 – 11y + 2) = (4c – d)(2c + 5d) = 2x (5y – 2) (y – 1) e) 24(x – y)2 – 13(x – y) – 2 Get students to rewrite the trinomial with B (for bracket) instead. = 24B2 – 13B – 2

-48 / \ +3 -16 = 24B2 + 3B – 16B – 2 = 3B(8B + 1) – 2(8B + 1) = (8B + 1) (3B – 2) (remind students to back substitute the “B” back) = (8(x-y) + 1)(3(x-y) – 2) = (8x – 8y + 1) (3x – 2y – 2) (simply the brackets) Practices MHR p.234 # 2ac, 3, 6, 7adefghj, 8b, 11c, 12, 17, 18

Polynomials Lesson 4 Factoring Special Quadratics and Special Cubes Eg1. Expand. a) (y + 7)(y – 7) b) ( )26k + c) (2x – 3)(2x + 3) d) ( )23 4h− In fact, with a quadratic starting and ending with a perfect square, chances are, you are working with a special quadratic. It may be a perfect trinomial or a difference of squares. Perfect Trinomials: ( )2 2 22a b a ab b+ = + + or ( )22 22a ab b a b+ + = +

( )2 2 22a b a ab b− = − + or ( )22 22a ab b a b− + = − Difference of Squares: ( )( ) 2 2a b a b a b− + = − or ( )( )2 2a b a b a b− = − +

Eg2. Factor the following. a) 2 10 25a a+ + b) 49x2 – 121

Eg3. Factor the following. a) 2 24 12 9a ab b+ + b) 50m2 – 32n4 c) ( ) ( )2 25 2 3 4m m− − − d) 225x4 – (2x + 1)2 We can also factor the difference of cubes and the sum of cubes. ( )( )3 3 2 2a b a b a ab b− = − + + and ( )( )3 3 2 2a b a b a ab b+ = + − +

Eg4. Factor completely. a) x3 – 125 b) 27 – 64k3 c) 32a5b3 + 4a2b3 Practices MHR p.247 # 1, (2, 3)cd, 4, 5cdgh, 6acef, 7bcef, 8ac, 14, 20, 21 Worksheet # 1 – 3

Worksheet 1. Factor the following difference of squares. a) (2a+b)2 – 81 b) 81a2 – (3a + b)2 c) 4(2x – y)2 – 25z2 d) (2x – 1)2 – (7x + 4)2 2. Factor the following trinomials. a) x4 – 13x2 + 36 b) a4 – 17a2 + 16 c) y4 – 5y2 – 36 3. Factor the following. a) 273 −x b) 643 +y c) 33 278 yx −

d) 3192 3 +u e) 664 x− f) ( ) 4025 3 −−x

g) 433 2502 zzyx + h) ( )33 5227 −− aa i) nn ba 33 + Answers 1. a) (2a + b – 9)(2a + b + 9) b) [9a – (3a + b)][9a + (3a + b)] = [6a – b][12a + b] c) [2(2x – y) – 5z][2(2x – y) + 5z] = (4x – 2y – 5z)(4x – 2y + 5z) d) (-5x – 5)(9x + 3) = -15(x + 1)(3x + 1) 2. a) (x2 – 4)(x2 – 9) = (x – 2)(x + 2)(x – 3)(x+ 3) b) (a2 – 1)(a2 – 16) = (a – 1)(a + 1)(a – 4)(a + 4) c) (y2 + 4)(y2 – 9) = (y2 + 4)(y – 3)(y + 3) 3. a) ( )( )933 2 ++− xxx b) ( )( )1644 2 +−+ yyy c) ( )( )22 96432 yxyxyx ++− d) ( ) ( )( )14161431643 23 +−+=+ uuuu e) ( )( ) ( )( )( )( )2233 24224288 xxxxxxxx +−+++−=+−

f) ( )[ ] ( )[ ]( ) ( )[ ] ( )( )42454222225825 223 +−−=+−+−−−=−− xxxxxxx g) ( ) ( )( )222333 255521252 zxyzyxzxyzzyxz +−+=+

h) ( )3 (2 5)a a− − { ( ) ( )229 3 2 5 2 5a a a a+ − + + } ( )( )25 19 35 25a a a= + − +

i) ( )na 3+ ( )nb 3 ( )( )2 2n n n n n na b a a b b= + − +