ALGEBRA Basic Algebra Skills Expansion using Distributive Law
ALGEBRA
Basic Algebra Skills Expansion using Distributive Law Ø a(b + c) = ab + ac. Example 1: 2 x(3 x + 1) = 6 x² + 2 x Ø a(b – c) = ab – ac. Example 2: 3(2 x - 1) == 6 x 6 x -- 33 Expansion means removing brackets http: //www. mathsisfun. com/algebra/expanding. html
Basic Algebra Skills Expansion using Distributive Law Ø (a + b)(c + d) = ac + ad + bc + bd Ø (a + b)(c – d) = ac – ad + bc – bd Ø (a – b)(c – d) = ac – ad – bc + bd http: //www. mathsisfun. com/algebra/expanding. html
Smiling Man ☺ Used to do expansion
Example 1 ☺ Expand the following Expression (x+2) (x+3) =
(x+2) (x+3) = x 2 + ☺ [x multiplied by x equals x 2 ]
☺ (x+2) (x+3) = 2 x + 3 x [x multiplied by 3 equals 3 x]
☺ (x+2) (x+3) = 2 x + 3 x + 2 x [2 multiplied by x equals 2 x]
☺ (x+2) (x+3) = 2 x + 3 x + 2 x + 6 [2 multiplied by 3 equals 6]
(x+2) (x+3) ☺ If done correctly you should see: • Two top arcs form the eyebrows. • The outer brackets form the ears • Two lower arcs and the inner brackets form the nose and mouth.
☺ (x+2) (x+3) = x 2 + 3 x + 2 x + 6 [simplify by adding 2 x + 3 x]
☺ (x+2) (x+3) = x 2 + 5 x + 6 [simplify by adding 2 x + 3 x]
Example 2 ☺ Expand the following Expression (x+2) (x-3) =
(x+2) (x-3) = x 2 + ☺ [x multiplied by x equals x 2 ]
☺ (x+2) (x-3) = 2 x - 3 x [x multiplied by -3 equals -3 x]
☺ (x+2) (x-3) = 2 x - 3 x + 2 x [2 multiplied by x equals 2 x]
☺ (x+2) (x-3) = 2 x - 3 x + 2 x + 6 [2 multiplied by 3 equals 6]
(x+2) (x-3) ☺ If done correctly you should see: • Two top arcs form the eyebrows. • The outer brackets form the ears • Two lower arcs and the inner brackets form the nose and mouth.
☺ (x+2) (x-3) = x 2 - 3 x + 2 x + 6 [simplify by adding 2 x - 3 x]
☺ (x+2) (x-3) = x 2 - x + 6 [simplify by adding 2 x + 3 x]
☺ Example 3 Expand the following Expression (x-2) (x-3) =
(x-2) (x-3) = x 2 + ☺ [x multiplied by x equals x 2 ]
☺ (x-2) (x-3) = 2 x - 3 x [x multiplied by -3 equals -3 x]
☺ (x-2) (x-3) = 2 x - 3 x - 2 x [-2 multiplied by x equals -2 x]
☺ (x-2) (x-3) = 2 x - 3 x - 2 x + 6 [-2 multiplied by -3 equals 6]
(x-2) (x-3) ☺ If done correctly you should see: • Two top arcs form the eyebrows. • The outer brackets form the ears • Two lower arcs and the inner brackets form the nose and mouth.
☺ (x-2) (x-3) = x 2 - 3 x - 2 x + 6 [simplify by adding -2 x - 3 x]
☺ (x+2) (x+3) = x 2 - 5 x + 6 [simplify by adding -2 x - 3 x]
Example 4 ☺ Expand the following Expression (x+2) (3 -x) =
(x + 2) (3 - x)= 3 x ☺ [x multiplied by 3 equals 3 x ]
☺ (x+2) (3 -x) = 3 x – 2 x [x multiplied by -x equals -x 2]
☺ (x+2) (3 -x) = 3 x – 2 x +6 [2 multiplied by 3 equals 6]
☺ (x+2) (3 - x) = 3 x – 2 x + 6 - 2 x [2 multiplied by -x equals -2 x]
(x+2) (3 -x) ☺ If done correctly you should see: • Two top arcs form the eyebrows. • The outer brackets form the ears • Two lower arcs and the inner brackets form the nose and mouth.
☺ (x+2) (3 -x) = -x 2 + 3 x - 2 x + 6 [simplify by adding 3 x - 2 x]
☺ (x+2) (3 -x) = -x 2 + x + 6 [simplify by adding -2 x - 3 x]
☺ Is there anybody prepared to demonstrate the "smiling man" technique on the board?
Basic Algebra Skills Factorising (add brackets) Expand This is the inverse (opposite) of expanding brackets 2 x(3 x + 1) = 6 x² + 2 x Factorise v 2 terms Example 1: Factorise completely 4 x 2 + 12 x Method: HCF 4 4 x² + 12 x x² + 3 x x+3 Stop! No more common factor s x Answer: 4 x(x + 3)
Basic Algebra Skills Factorising (adding brackets) v 2 terms Example 2: Factorise completely 24 a + 16 Method: HCF 8 24 a + 16 Stop! No more common factor s 3 a + 2 Answer: 8(3 a + 2)
Basic Algebra Skills Factorising (adding brackets) v 2 terms Example 3: Factorise completely x² - x Method: HCF x² - x x-1 Answer: x(x - 1) Stop! No more common factor s x
Basic Algebra Skills Factorising (adding brackets) v 2 terms Example 4: Factorise y(b + 2) – 3(b + 2) Method: HCF y(b + 2) – 3(b + 2) y-3 Answer: (b + 2)(y - 3) Stop! No more common factor s (b + 2)
Exercise 2 1. Factorise the following expressions. (a) 2 m + 4 (b) 15 – 3 b (c) x² + 3 x (d) 12 ab – 20 a (e) 10 xy + 5 x³ (f) 16 y³ + 8 xy – 14 y (g) 14 m – 28 m² + 35
Exercise 3 1. Factorise the following expressions. (a) 56 + 16 y (b) 21 – 28 v (c) – 36 v + 24 u (d) – 25 s – 20 t (e) 2 x + 3 x² (f) 4 p² – 12 py
Basic Algebra Skills Factorising v 4 terms (by grouping) Example 1: Factorise fully, ac + ad + 3 c + 3 e Method: HCF a ac + ad 2 3 + 3 c + 3 e c+d c+e a( c + d) 3(c + e) Answer: a(c + d) + 3(c + e) 1 Stop! No more common factor s 1 Grouping: (ac + ad)(+ 3 c + 3 e) 2
Basic Algebra Skills Factorising v 4 terms (by grouping) Example 2: Factorise fully, 2 uv – 6 us + tv – 3 st Method: HCF 2 2 uv - 6 us u uv – 3 us v – 3 s 2 u( v – 3 s) 2 t + tv – 3 st v – 3 s 1 2 Stop! No more common factor s 1 Grouping: (2 uv – 6 us) (+ tv – 3 st) t( v – 3 s) 2 u(v – 3 s) + t(v -3 s) Ans: (v – 3 s)(2 u+ t)
Basic Algebra Skills Factorising v 4 terms (by grouping) Example 3: Factorise fully, mp + 2 mq – 3 np – 6 nq Method: HCF 1 m mp+2 mq p + 2 q m( p + 2 q) Grouping: (mp + 2 mq) (- 3 np – 6 nq) 2 - -3 np – 6 nq 3 +3 np + 6 nq n +np + 2 nq p + 2 q -3 n( p + 2 q) 1 2 m(p + 2 q) -3 n(p +2 q) Answer: (p + 2 q)(m - 3 n)
Basic Algebra Skills Factorising v 4 terms (by grouping) Example 4: Factorise fully, hx + ky - hy - kx Method: HCF 1 h hx-hy x-y Arrange them with common factors first = hx – hy - kx + ky Grouping: (hx - hy) (- kx +ky) 2 - -kx + ky k +kx-ky x-y 1 2 h(x - y) -k(x - y) h(x-y) -k(x-y) Answer: (x - y)(h - k)
Exercise 4 1. Factorise the following expressions. (a) 3 uv + 9 u – 2 v – 6 (b) 10 pq – 5 p + 6 q – 3 (c) 4 ac + 8 bc – 3 a – 6 b (d) mx – my + nx – ny
Basic Algebra Skills Factorising v. Perfect squares Formula: x² - y² = (x + y) (x – y) Example 1: Factorise completely x² - 4 Answer: (x + 2) (x – 2)
Basic Algebra Skills Factorising v. Perfect squares Formula: x² - y² = (x + y) (x – y) Example 2: Factorise completely x² - 64 Answer: (x + 8) (x – 8)
Basic Algebra Skills Factorising v. Perfect squares Formula: x² - y² = (x + y) (x – y) Example 3: Factorise completely 25 x² - 9 y² Answer: (5 x + 3 y) (5 x – 3 y)
Basic Algebra Skills Factorising v. Perfect squares Formula: x² - y² = (x + y) (x – y) Example 4: Factorise completely 4 - 36 z² Answer: (2 + 6 z) (2 - 6 z)
Basic Algebra Skills Factorising v. Perfect squares Formula: x² - y² = (x + y) (x – y) Example 5: Evaluate 36² - 4² Answer: (36 + 4) (36 – 4) = 40 x 32 = 1280
Basic Algebra Skills Factorising v. Perfect squares Formula: x² - y² = (x + y) (x – y) Example 6: Evaluate 10² - 5² Answer: (10 + 5) (10 – 5) = 15 x 5 = 75
Exercise 1. Factorise the following expressions (a) m² – 4 (b) n² – 1 2. Factorise the following expressions (a) 25 n² – 4 (b) h² – 121 3. Evaluate the following expressions (a) 64² − 36² (b) 6. 5² – 3. 5²
Basic Algebra Skills Factorising v. Quadratic Expression (ax 2 + bx + c, where a, b, c are constant) Example 1: Factorise the quadratic expression: x 2 + 4 x + 3 Possibilities 1 x 3 Method: x 3 3 x x 1 x x 2 3 4 x Answer: (x + 3)(x + 1)
Basic Algebra Skills Factorising v. Quadratic Expression (ax 2 + bx + c, where a, b, c are constant) Example 2: Factorise x 2 - 8 x + 12 Method: x -2 -2 x x -6 -6 x x 2 12 -8 x Possibilities 2 x 6 1 x 12 3 x 4 -1 x -12 -2 x -6 -3 x -4 Answer: (x - 2)(x - 6)
Basic Algebra Skills Factorising v. Quadratic Expression (ax 2 + bx + c, where a, b, c are constant) Example 3: Factorise the quadratic expression: x 2 + x -12 Method: x -3 -3 x x 4 4 x x 2 -12 x Possibilities 2 x -6 1 x 12 3 x -4 -1 x 12 -2 x 6 -3 x 4 Answer: (x - 3)(x +4)
Basic Algebra Skills Factorising v. Quadratic Expression (ax 2 + bx + c, where a, b, c are constant) Example 4: Factorise the quadratic expression: x 2 - x - 2 Possibilities 1 x -2 -2 x 1 Method: x 1 x x -2 -2 x x 2 -2 -x Answer: (x + 1)(x - 2)
Exercise 1 1. Factorise the following expressions (a) x² + 8 x + 7 (b) y² + 7 y + 12 2. Factorise the following expressions (a) x² – 11 x + 18 (b) x² – 3 x - 18
Exercise 2 1. Factorise the following expressions (a) m² + 12 m + 35 (b) 2 x² + 23 x + 15 2. Factorise the following expressions (a) 5 y² + 4 y - 12 (b) 2 x² + 9 xy + 9 y²
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