Polynomial is a general description on any algebraic expression with 1 term or more. To add or subtract polynomials, we combine like terms.

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1 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) x + 3 b) -5a b + a 5 c) p + 3p 3 q p d) -4yx + 3 x y + 5x y

2 Eg. Simplify. a) (3x 6x + 7) (7x x + 9) b) (a b b ) + (3b a + a ) c) -8(k 3) + 11(k ) d) 3a (-a 3 + b 4a b) b(a 5) To simplify expressions with a multiplication of two monomials, we use BEDMAS. Eg1. Simplify. a) (3x)( 5x ) 3 b) (6p q)( 3pq ) c) 4 5 (5m n )( mn ) 3 8 ( 10m n ) 3 Practices: Worksheet # 1cf, gh, 3gh, 4bf, 5df, 9ef, 10ce, 11ef, 18bf, 1ad, 5ad, 7bf, 31be

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5 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 cm. Find its final area. using algebra tiles: using distributive property:

6 Eg. Use algebra tiles to determine the product (x + 3) (x 1). Eventually, multiplying polynomials would be more efficient with the distributive property. Eg3. Expand x ( x + ). Using similar idea, we can distributive a binomial into another binomial. Eg4. Expand ( )( x + 3) x. The expansion above is so frequently used that there is an acronym for it. FOIL: Eg5. Expand and simplify. a) (x 3)(x + 5) b) (4a 3b)

7 To multiply a binomial by a polynomial, we can extend the idea of FOIL. Eg6. Expand. a) (3x 5) (5x xy + y ) b) (4x 1) (3x + x 5) c) (x + 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 cm. Practices: MHR p.09 # 1adf,, 3df, 4, 5bd, 6ade, 7, 10, 11, 13, 15, 17, 19 Polynomials

8 Lesson 5. 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 1y. a) State the common factor for 18x and 1y. b) Factor the expression 18x 1y. 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. Eg. 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) x 8 b) 4x 3 y x y c) 55J Δk 10kΔJ

9 Idea: Factor out GCF for all coefficients, and use lowest exponent for each variable. Eg4. Factor completely. a) 4x (x 4) 4xy(x 4) b) a 3 b (c d) + a b 3 (d c) Eg5. Factor by first grouping similiar terms together. a) y + 8xy+ 16x+ y b) cx 8x + 7c 8 c) y + 8xy+ y+ 16x Eg6. Joey the Little Monster went trick or treat on the Halloween night. He got 1 Snickers, 4 Starburst, 8 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.0 # ce, 4be, 5, 6, 7abc Worksheet #1(all), (all)

10 MHR p.0 # 10, 1, 13, 15, 16, 18, 19ab Worksheet 1. Factor the following. 3y y 4 y a) ( ) + ( ) b) 5aa ( 4) + ( 4 a) c) m( x y) 5( y x) d) 7wx ( + w) 10( w+ x) e) ( x+ 4) x ( 4+ x) 5 f) ( m 5n) a ( 5n m) bc g) ma ( b) 3na ( b) 7( a b) h) x ( 3a b) + 5x( 3a b) 9( b 3a). Factor the following by grouping. a) x + 3x+ xy+ 3y b) 5am + a + 10bm + b 3 c) x + x + x+ 1 d) 3x 6xy+ 5x 10y e) 5m + 10mn 3m 6n f) a 6ab 3a+ 9b 6 4 g) y + y 10y h) x + x x 1 Answers 1. a) ( 3y+ 4)( y ) b) ( 5a )( a 4) c) ( m+ 5)( x y) d) ( 7w 10)( w+ x) e) ( x+ 4)( x 5) f) ( m 5n)( a + bc) g) ( m 3n 7)( a b) h) ( x + 5x+ 9)( 3a b). a) ( x+ 3)( x+ y) b) ( a+ b)( 5m+ 1) c) ( x + 1)( x+ 1) d) ( 3x+ 5)( x y) e) ( 5m 3)( m+ n) f) ( a 3)( a 3b)

11 4 3 g) ( y 5)( y + 1) h) ( x 1)( x + 1) Polynomials Lesson 3a Factoring Trinomials Eg1. Expand by FOIL. a) (x + 3)(x + 5) b) (k )(k + 7) In general, expanding (x + a) (x + b) results in x + 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;. Check if the sum/difference of these two numbers would equal the middle term; 3. Use these two numbers to factor the trinomial. Eg. Factor the given trinomials. a) x + 11x + 4 b) a + 5a 6 c) 10y 0y 150 d) 6 60 ab ab a

12 Remember: Always check for common factor before factorizing the trinomial. Eg3. Factor the following polynomials. a) 3k 4 + 3k 36 b) 5a b 65ab 150 c) 14m + 5mn n d) (x + y) 8(x + y) + 15 Eg4. For which integral values of k can the trinomial be factored? x + kx 6 Practices Worksheet # (3, 5, 7)behk, 13ac, 14af, 18, 19abef, 0cd, 1cdf MHR p.35 # 8a, 1(all)

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14 Polynomials Lesson 3b Factoring Trinomials with Leading Coefficient Eg1. Factor the following. a) 1 5c + 6c b) x + xy 3y / \ / \ c 3c -y +3y = (1 c) (1 3c) = (x y)(x + 3y) c) a b + 5ab 6 d) k k + 36k / \ +6-1 Factor out the k first! = (? + 6 ) (? 1 ) = k (k + 9k + 18) / \ = (ab + 6) (ab 1) = k (k + 3) (k + 6) When we are to factor trinomials with a leading coefficient: ax + bx + c We use the method of decomposition : It begins by multiplying the leading coefficient (a) to the last term (c). Eg. Factor a) x + 7x + 6 b) 3x 10x use the same method as in (a) / \ ans: (3x + ) (x 4) = x + 3x + 4x + 6 (emphasize we only decompose the middle term (7x) into two pieces) as we don t change the front (x ) or the back (6) = x + 3x + 4x + 6 (split the 4 terms into groups then factor by grouping) draw a line to indicate the two groups = x (x + 3) + (x + 3) (identify the common factor for the first terms & the last terms) = (x + 3) (x + ) (factor out x + 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.

15 a) 4x + 4x 3 b) k 14k + 1 You can get students to try these now. Remind them to identify the common factor if there are any! = (x + 3) (x 1) = (k 7k + 6) = (k 1)(k 6) c) 10xy xy + 4x d) 8c + 18cd 5d = x (5y 11y + ) = (4c d)(c + 5d) = x (5y ) (y 1) e) 4(x y) 13(x y) Get students to rewrite the trinomial with B (for bracket) instead. = 4B 13B -48 / \ = 4B + 3B 16B = 3B(8B + 1) (8B + 1) = (8B + 1) (3B ) (remind students to back substitute the B back) = (8(x-y) + 1)(3(x-y) ) = (8x 8y + 1) (3x y ) (simply the brackets) Practices MHR p.34 # ac, 3, 6, 7adefghj, 8b, 11c, 1, 17, 18

16 Polynomials Lesson 4 Factoring Special Quadratics and Special Cubes Eg1. Expand. a) (y + 7)(y 7) b) ( k + 6) c) (x 3)(x + 3) d) ( 3h 4) 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: Difference of Squares: ( ) a+ b = a + ab+ b or a + ab+ b = ( a+ b) ( ) a b = a ab+ b or a ab+ b = ( a b) ( a b)( a b) a b + = or a b = ( a b)( a+ b) Eg. Factor the following. a) a + 10a+ 5 b) 49x 11

17 Eg3. Factor the following. a) 4a + 1ab+ 9b b) 50m 3n 4 c) ( 5m ) ( 3m 4) d) 5x 4 (x + 1) We can also factor the difference of cubes and the sum of cubes. = ( )( + + ) and a 3 + b 3 = ( a+ b)( a ab+ b ) a b a b a ab b 3 3 Eg4. Factor completely. a) x 3 15 b) 7 64k 3 c) 3a 5 b 3 + 4a b 3 Practices MHR p.47 # 1, (, 3)cd, 4, 5cdgh, 6acef, 7bcef, 8ac, 14, 0, 1 Worksheet # 1 3

18 Worksheet 1. Factor the following difference of squares. a) (a+b) 81 b) 81a (3a + b) c) 4(x y) 5z d) (x 1) (7x + 4). Factor the following trinomials. a) x 4 13x + 36 b) a 4 17a + 16 c) y 4 5y Factor the following. a) x 3 7 b) y c) d) 19u e) g) z 6 64 x x y z + 50 h) 7 3 ( a 5 ) x 7 y 3 f) 5( x ) 40 a i) a 3n + b 3n Answers 1. a) (a + b 9)(a + b + 9) b) [9a (3a + b)][9a + (3a + b)] = [6a b][1a + b] c) [(x y) 5z][(x y) + 5z] = (4x y 5z)(4x y + 5z) d) (-5x 5)(9x + 3) = -15(x + 1)(3x + 1). a) (x 4)(x 9) = (x )(x + )(x 3)(x+ 3) b) (a 1)(a 16) = (a 1)(a + 1)(a 4)(a + 4) c) (y + 4)(y 9) = (y + 4)(y 3)(y + 3) 3. a) ( x 3)( x + 3x + 9) b) ( y + 4)( y 4y + 16) c) ( x 3y)( 4x + 6xy + 9y ) 3 d) 3( 64u + 1) = 3( 4u + 1)( 16u 4u + 1) 3 3 e) ( 8 x )( 8 + x ) = ( x)( 4 + x + x )( + x)( 4 x + x ) 3 [ 8] = 5 ( x ) f) 5 ( x ) [ + 4] = 5( x 4)( x x + 4) [ ]( x ) + ( x ) g) z ( x y + 15z ) = z( xy + 5z)( x y 5xyz + 5z ) h) ( 3 a (a 5) ){ 9a + 3a( a 5) + ( a+ 5) } = ( a+ 5)( 19a 35a+ 5) n i) ( a ) 3 n + ( b ) 3 = ( a n + b n )( a n a n b n + b n )