Revision Algebra II

1. Revision Algebra II Factorisation of Binomials, Trinomials, Sum & Difference of Two Cubics By I Porter

2. Factors • A factor of a term or number divides into that term or number, without a remainder. • For examples, the factors of • 18 are 1, 2, 3, 6, 9 and 18. • 6x2 are 1, 2, 3, 6, x, 2x, 3x, 6x, x2, 2x2, 3x2 and 6x2. • 4xy are 1, 2, 4, x, y, 2x, 2y, 4x, 4y, xy, 2xy and 4xy. • The highest common factor (HCF) of 4xy and 6x2 is 2x. The product of a(b + c) is ab + ac and the product of (a + b)(c + d) is ac + ad + bc + bd, we say, these two results are written in expanded form. When they are written in the form a(b + c) and (a + b)(c + d), we say they are in factored form. Factorisation is the process whereby an expression in expanded form, such as ab + ac, is changed to factored form, such as a(b + c).

3. = x(a + b) + y(a+ b) = (a + b) Factorising by Grouping Algebraic expressions with 4 terms do not have a factor common to every term. Such expressions can often be factorised by first grouping the terms in pairs. ax+ bx + ay +by (x + y) Examples: Fully factorise the following a) a(b - 3) + 6(b - 3) b) ac + ad + bc + bd c) 4x2 - 4xy + x - y = a(b - 3) + 6(b - 3) = ac + ad + bc + bd = 4x2 - 4xy + x - y = (b - 3) (a + 6) = a(c + d) + b(c + d) = 4x(x - y) + 1(x - y) = (c + d) (a + b) =(x - y) (4x + 1)

4. More Examples: [negative signs as a factor] e) 6x2 - 4x - 3xy + 2y d) mn + 4n - m - 4 = 2 x3x2 -2 x 2x-3xy+2y = mn + 4n-(m+ 4) = 2x(3x - 2)- y(3x -2) = n(m + 4) - 1(m + 4) = (3x - 2)(2x -y) = (m + 4) (n - 1) Exercise: Fully factorise the following a) ab + 5a + 2b + 10 b) xy - 3x + 2y - 6 c) 3c2 - 2cd + 3cd2 - 2d3 = (a + 2)(b + 5) = (x + 2)(y - 3) = (c + d2)(3c - 2d) d) ab + 2b - 3a - 6 e) 2x2 - 6x - xy + 3y f) 4c2 + 12c - cd - 3d = (a + 2)(b - 3) = (2x - y)(x - 3) = (4c - d)(c + 3)

5. = (a - b)(a + b) Factorising the difference of 2 squares: a2 - b2 Examples: Factorise. b) x2 - 9 c) 9x2 - 25 a) n2 - p2 = (x)2 - (3)2 = (3x)2 - (5)2 = (n - p) (n + p) = (x - 3) (x + 3) = (3x - 5) (3x + 5) d) 4x2 - y2 e) 5x2 - 45 f) 32x2 - 18y2 = (2x)2 - (y)2 = 5(x2 - 9) = 2(16x2 - 9y2) = (2x - y) (2x + y) = 5[(x)2 - (3)2] = 2[(4x)2 - (3y)2] = 5(x - 3) (x + 3) = 2(4x - 3y) (4x + 3y)

6. Examples: Factorise a) x3 - 8 b) 8x3 + 27 Exercise: Factorise a) x2 - 36 b) 4x2 - 49 c) 9x2 - 25y2 = (x - 6)(x + 6) = (2x - 7)(2x + 7) = (3x - 5y)(3x + 5y) d) 8x2 - 50 e) 18x2 - 72y2 = 2(2x - 5)(2x + 5) = 18(x - 2y)(x + 2y) Must know for HSC Factorisation of CUBICS: a3 - b3 and a3 + b3 a3 - b3 = (a - b)(a2+ab + b2) a3 + b3 = (a + b)(a2-ab + b2) = x3 - 23 =(2x)3 + (3)3 = (x - 2)(x2+ 2x + 22) = (2x + 3)(4x2- 6x + 32) = (x - 2)(x2+ 2x + 4) = (2x + 3)(4x2- 6x + 9)

7. Exercise: Factorise a) x3 - 125 b) 8x3 + y3 c) 9x3 - 72y3 = (x - 5)(x2+ 5x + 25) = (2x + y)(4x2- 2xy + y2) = 9(x - 2y)(x2+ 2xy + 2y2) Factoring quadratic trinomials: ax2 ± bx ± c • A trinomial has 3 terms. • A quadratic trinomial is an expression with 3 terms where the highest power of • a term is 2. Sum and Product Method. To factorise a trinomial of the form x2 ± bx ± c (where b and c are non-zero integers) look for two numbers P and Q such that: * Their sum is b : (P + Q = b) * Their product is c : (PQ = c) [ The first step to start may be to write out the factors of c in a systematic process.]

8. Examples: Factorise the following: Sum = +2 Product = -24 a) x2 + 7x +12 c) x2 + 2x - 24 Sum = +7 Product = +12 Factors of -24 -24 = -1 x 24 = -2 x 12 = -3 x 8 = -4 x 6 Factor sums -1 + 24 = +23 -2 + 12 = +10 -3 + 8 = +5 -4 + 6 = +2 Factors of +12 12 = 1 x 12 = 2 x 6 = 3 x 4 Factor sums 1 + 12 = +13 2 + 6 = +8 3 + 4 = +7 Why not the (-) factors? Sum and Product are both (+) Why do the factors have different signs? Sum (+) and Product (-), largest number is (+) So, x2 + 2x - 24 = (x - 4)(x + 6) So, x2 + 7x +12 = (x + 3)(x + 4) Sum = -3 Product = -40 Sum = -8 Product = +12 b) x2 - 8x +12 d) x2 - 3x - 40 Factors of -40 -40 = 1 x -40 = 2 x -20 = 4 x -10 = 5 x -8 Factor sums 1 + -40 = -39 2 + -20 = -18 4 + -10 = -6 5 + -8 = -3 Factors of +12 12 = -1 x -12 = -2 x -6 = -3 x -4 Factor sums -1 + -12 = -13 -2 + -6 = -8 -3 + -4 = -7 Why not the (+) factors? Sum (-) and Product (+) both are (-) Why do the factors have different signs? Sum (-) and Product (-), largest number is (-) So, x2 - 8x +12 = (x - 2)(x - 6) So, x2 - 3x - 40 = (x + 5)(x - 8)

9. Examples: Factorise a) 2x2 - 10x - 28 b) 4x2 - 20x + 24 Exercise: Factorise a) x2 - 9x + 20 b) x2 + 10x + 24 c) x2 - 8x - 48 = (x - 4)(x - 5) = (x + 4)(x + 6) = (x + 4)(x - 12) d) x2 + 2x - 48 e) x2 - 15x + 54 e) x2 + 12x - 64 = (x + 8)(x - 6) = (x - 9)(x - 6) = (x + 16)(x - 4) = 2(x2 - 5x - 14) Sum = -5 Product = -14 = 4(x2 - 5x + 6) Sum = -5 Product = +6 Factors of -14 -14 = 1 x -14 = 2 x -7 Factor sums 1 + -14 = -13 2 + -7 = -5 Factors of +6 +6 = -1 x -6 = -2 x -3 Factor sums -1 + -6 = -7 -2 + -3 = -5 So, 2x2 - 10x - 28 = 2(x + 2)(x - 7) So, 4x2 - 20x - 28 = 4(x - 2)(x - 3)

10. Harder factorisation of quadratic trinomials: ax2 + bx + c Examples: Factorise a) 3x2 - 2x - 8 b) 6x2 + 13x - 5 Exercise: Factorise a) 7x2 - 63x + 140 b) 12x2 - 12x - 72 c) 9x2 + 36x + 27 = 7(x - 4)(x - 5) = 12(x + 2)(x - 3) = 9(x + 1)(x + 3) [ Variation of sum + product method] Step 1: multipy first and last term. 6 x -5 = -30 Step 2: Look for two numbers that have a sum of +13 and product of  -30. (I.e. -2 and +15) Step 3: Split the middle term using the number found in step 2 6x2 - 2x + 15x - 5 Step 4: Factorise. Step 1: multipy first and last term. 3 x -8 = -24 Step 2: Look for two numbers that have a sum of -2 and product of  -24. (I.e. -6 and +4) Step 3: Split the middle term using the number found in step 2 3x2 - 6x + 4x - 8 Step 4: Factorise. 6x2 + 13x - 5 = 6x2 - 2x + 15x - 5 3x2 - 2x - 8 = 3x2 - 6x + 4x - 8 = 2x(3x - 1) + 5(3x - 1) = 3x(x - 2) + 4(x - 2) = (3x - 1) (2x + 5) = (x - 2) (3x + 4) If you can use the CROSS method successfully, KEEP USING IT!

11. Exercise: Factorise a) 2x2 + 5x + 2 b) 5x2 - 17x - 12 c) 6x2 - 7x + 1 (2x + 1)(x + 2) (5x + 3)(x - 4) (6x - 1)(x - 1) d) 4x2 + 7x - 2 e) 6x2 - 7x - 20 f) 8 + 10x - 3x2 (4x - 1)(x + 2) (2x - 5)(3x + 4) (2 + 3x)(4 - x)