Transcript Factorising.ppt

Factorising
Identifying and extracting common
factors from groups of terms
JR/2008
Factorising
 A factor is a common part of two or more terms making
up an expression
eg. 3x + 3y has two terms. The number 3 is common to
both, this can be extracted as a common factor thus:3x + 3y becomes 3 (x + y )
 The simple method is
1. Find HCF (highest common factor) of all the terms in
the expression.
2. HCF moved outside of brackets
3. Terms left inside the bracket are the original
expression terms divided by the HCF.
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Factorising
 Finding the HCF
eg. find HCF of 3m2np3, 6m3n2p2, 24m3p4
First consider numbers: Factor of 3, 6, 24 = 3
Then look at terms: m and p are in all three terms
lowest powers of m and p are: m2 p2
The HCF is therefore 3 m2 p2
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Factorising
 Factorising by grouping
eg. ax + ay + bx + by
The above can be grouped as: (ax + ay) + (bx + by)
Factorise each bracket: a (x + y) + b (x + y)
Note: x + y is common: giving (a + b)(x + y)
Thus ax + ay + bx + by can be written as (a + b)(x + y)
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Factorising
 Examples
1. Find factors of zx + px
Solution:
x (z + p)
2. Find factors of z2x - 3zx2
Solution: the HCF of z2x and 3zx2 is zx
Thus the factors of z2x - 3zx2 are zx (z – 3x)
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Factorising
3. Find factors of 3z4x + 12z3x2 - 6z2x3
Solution: the HCF of 3z4x, 12z3x2, 6z2x3 is 3 z2x
Thus 3z4x + 12z3x2 - 6z2x3 becomes 3 z2x(z2 + 4zx – 2x2)
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Factorising
4. Find factors of np + mp – zn – zm (by grouping)
Solution: grouping gives: (np + mp) – (zn + zm)
Take care with the signs !!!!
Factorise each: p(n + m) – z(n + m)
Thus np + mp – zn – zm gives (p – z)(n + m)
Alternative method: grouping (np – zn) + (mp – zm)
gives: n(p – z) + m(p – z) again giving (n + m)(p – z)
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Factorising
Try:
1. m2n – 2mn2
2. 3x4y + 9x3y2 – 6x2y3
3. ac + bc + dc
x
x2
x3
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Factorising
Try: (by grouping first)
4. az + by + bz + ay
5. mp + np – mq - nq
6. a2c2 + acd + acd + d2
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Factorising
Try: (by grouping first)
7. 2pz – 4ps + bz – 2sb
8. mn(3x – 1) – pq(3x – 1)
9. k2e2 – mne – k2e + mn
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Factorising
Solutions
1. mn(m – 2n)
2. 3x2y(x2 + 3xy – 2y2)
3. c (a + b + d )
x
x
x2
4. (a + b)(z + y)
5. (m + n)(p – q)
6. (ac +d)2
7. 2p(z – 2s) + b(z – 2s) = (2p + b)(z – 2s)
8. (mn – pq)(3x – 1)
9. (k2e – mn)(e – 1)
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