Algebraic Fractions Simplification Techniques
Explore how to factorize and simplify algebraic fractions using common techniques such as factorization, canceling common factors, and multiplying/dividing fractions. The process involves identifying factors, canceling where possible, and performing operations to simplify expressions. Checkpoints and answers are provided to reinforce understanding and practice. Mastering these methods will aid in solving complex algebraic problems efficiently and accurately.
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Presentation Transcript
Starter Factorise the following: 8x - 32x = 8x(x 4) x + 5x 14 = (x + 7)(x 2) 3x + 17x + 10 = (3x + 2)(x + 5)
Simplifying Algebraic Fractions To simplify algebraic fractions: 1) factorise the top and/or the bottom 2) cancel common factors Why can t we cancel this further? 3x + 6 3x = 3(x + 2) 3x = x + 2 x = b + 2a 3 4ab + 8a 12a = 4a(b + 2a) 12a 3
Checkpoint 7x 35 21 x 5 3 3x + 15x 27x x + 5 9 6x y + 18xy 21xy 2(x + 3) 7 x + 7x + 10 3x + 15 x + 2 3
Simplifying Algebraic Fractions To simplify algebraic fractions: 1) factorise the top and/or the bottom 2) cancel common factors 4x - 20 x - 13x + 40 = 4(x - 5) (x 5)(x 8) = _4_ (x 8) = (x + 4)(x 4) (x + 4)(x 5) = (x 4) (x - 5) x - 16 x - x - 20
Simplifying Algebraic Fractions 5x 25 10x 12x + 30 9x 20x - 40x 15x 8ab 16b 24ab 9cd - 24d 15d 14efg + 7ef 21efg x + 9x + 20 7x + 28 x - 3x - 88 x - 21x + 110 x - 49 x - 5x - 14 2x + 7x + 3 x - 9 5x - 3x - 14 x - 11x + 18 6x - 13x - 5 10 - 19x - 15
Answers x 5 2x 2x + 5 3x 4x - 8 3x a - 2 3a 3cd 8d 5 2g + 7f 3g x + 5 7 x + 8 x - 10 x + 7 x + 2 2x + 1 x - 3 5x + 7 x - 9 3x + 1 5x + 3
Multiplying Algebraic Fractions To multiply algebraic fractions: 1) cancel common factors 2) multiply the numerators, multiply the denominators 3 4a x 15b 5 16 = a x 3b 4 = 3ab 4 4 = n + 4 x 3(n 3) n 3 5(n + 4) = 3 5 n + 4 x 3n 9 n 3 5n + 20
Dividing Algebraic Fractions To divide algebraic fractions: 1) flip the second fraction and change to x 2) cancel common factors 3) multiply the numerators, multiply the denominators 5= _1_ x _5_ 3xy 7y _11_ 77y 9x y 15x = _11_ x 15x 9x y 77y 3x = _5_ 21xy 7 x - 4 x - 5x + 6 x + 4x 32 5x + 40 = x - 4 x 5x + 40 x + 4x 32 x - 5x + 6 = 5(x + 2) (x - 4)(x 3) = (x + 2)(x 2) x 5(x + 8) (x + 8)(x 4) (x 2)(x 3)
Adding and Subtracting Fractions 2 + 1 3 4 1) Find a common denominator: 3 x 4 = 12 2) Find equivalent fractions with that denominator: 2 = 8 3 12 1 = 3 4 12 3) Add (or subtract!) the numerators and put the answer over the same denominator: 8 + 3 = 11 12 12 12 4) Simplify if possible.
Adding and Subtracting Algebraic Fractions To simplify algebraic fractions: 1) Find a common denominator 2) Find equivalent fractions with that denominator 3) Add (or subtract!) the numerators and put the answer over the same denominator 4) Simplify if possible. 2x + x 3 4 = 8x + 3x 12 12 = 11x 12 4 - 5 x 2x = 8 - 5 2x 2x = 3 2x
Adding and Subtracting Algebraic Fractions To simplify algebraic fractions: 1) Find a common denominator 2) Find equivalent fractions with that denominator 3) Add (or subtract!) the numerators and put the answer over the same denominator 4) Simplify if possible. x + 1 + x 3 4 7 = 7(x + 1) + 4(x 3) 28 28 = 7x + 7 + 4x + 12 28 = 11x - 5 28
Adding and Subtracting Algebraic Fractions To simplify algebraic fractions: 1) Find a common denominator 2) Find equivalent fractions with that denominator 3) Add (or subtract!) the numerators and put the answer over the same denominator 4) Simplify if possible. _9_ - _8_ x 2 x + 4 = 9(x + 4) - (x 2)(x + 4) (x + 4)(x 2) 8(x 2) = 9x + 36 8x + 16 x + 2x - 8 = x + 52 x + 2x - 8
Algebraic Fractions Adding and Subtracting 1) x x 7 + 5 2) 3 3 8x + 4x 3) 2 2 5x + 1 - 3x 4) 3 4 7x + 3x 5) 3 5 7x - 8x 6) 5x x 12 + 3 7) 6 4 x + 3 + x - 2 8) 3 7 2x 5 - 4x + 1 9) x + 4 x - 2 _7_ - _9_ 10) x 1 x + 1 _3_ + _2_
Algebraic Fractions Multiply and Divide 1) 8v 5v 7 x 2 2) 5 4t 6t 3 3) 7j 7 6j _6_ x 5j 5 4) 8x 4 6x + 3 10x + 5 5) 8w 24w - 16w 21w 14 x ___4____ 6) 10b + 20 2b + 4b __3b__ __9__ 7) a + 6a + 5 a + a 12 a - a 6 x a + 3a 10 8) c - 5c + 4 c + c 2 c + 12c + 32 c - 3c 28 9) 3f + 23f 8 2f - f 3 2f - 11f + 12 x 5f + 42f + 16 10) 6g + 7g + 2 5g + 21g + 4 10g - 13g - 3 2g 17g + 8
