Distributive Laws

Algebraic Techniques, SMB-066

Expand and simplify `(2x+3)(4x^2-6x+9).` (2 marks)

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` 8x^3+27`

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`(2x+3)(4x^2-6x+9)`	`=8x^3-12x^2+18x+12x^2-18x+27`
	`= 8x^3+27`

Algebraic Techniques, SMB-066

Factorise the expression `-12y^2+18y-6` (2 marks)

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`6(2y-1)(1-y)\ \ text(or)\ \ 6(1-2y)(y-1)`

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`-12y^2+18y-6=-6(2y^2-3y+1)`

`-6(2y^2-3y+1)=>\ \ `	`text{P}`	`=2xx1=2`
	`text{S}`	`=-3`
	`text{F}`	`=-2,-1`

`-6(2y^2-3y+1)`	`=(-6(2y-1)(2y-2))/2`
	`=-6(2y-1)(y-1)`
	`=6(2y-1)(1-y)\ \ text(or)\ \ 6(1-2y)(y-1)`

Algebraic Techniques, SMB-065

Fully factorise the expression `6y^2+10y-4` (2 marks)

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`2(3y-1)(y+2)`

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`6y^2+10y-4=2(3y^2+5y-2)`

`2(3y^2+5y-2)=>\ \ `	`text{P}`	`=3xx-2=-6`
	`text{S}`	`=5`
	`text{F}`	`=-1,6`

`2(3y^2+5y-2)`	`=(2(3y-1)(3y+6))/3`
	`=2(3y-1)(y+2)`

Algebraic Techniques, SMB-064

Factorise the expression `2x^2+x-6` (2 marks)

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`(2x-3)(x+2)`

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`2x^2+x-6=>\ \ `	`text{P}`	`=2xx-6=-12`
	`text{S}`	`=1`
	`text{F}`	`=-3,4`

`2x^2+x-6`	`=((2x-3)(2x+4))/2`
	`=(2x-3)(x+2)`

Algebraic Techniques, SMB-063

Fully factorise the expression `3a^2+12ab-2a-8b` (2 marks)

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`(3a-2)(a+4b)`

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`3a^2+12ab-2a-8b`	`=3a(a+4b)-2(a+4b)`
	`=(3a-2)(a+4b)`

Algebraic Techniques, SMB-062

Fully factorise the expression `2x^2-6xy-3x+9y` (2 marks)

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`(2x-3)(x-3y)`

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`2x^2-6xy-3x+9y`	`=2x(x-3y)-3(x-3y)`
	`=(2x-3)(x-3y)`

Algebraic Techniques, SMB-061

Factorise the expression `5b^3-5b` (2 marks)

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`5b(b+1)(b-1)`

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`5b^3-5b`	`=5b(b^2-1)`
	`=5b(b+1)(b-1)`

Algebraic Techniques, SMB-060

Expand and simplify the expression `(a+b)^2-(a-b)^2` (2 marks)

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`4ab`

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`(a+b)^2-(a-b)^2`	`=a^2+2ab+b^2-(a^2-2ab+b^2)`
	`=a^2+2ab+b^2-a^2+2ab-b^2`
	`=4ab`

Algebraic Techniques, SMB-059

Expand and simplify the expression `(4-7x)^2` (2 marks)

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`49x^2-56x+16`

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`(4-7x)^2`	`=(4-7x)(4-7x)`
	`=16-28x-28x+49x^2`
	`=49x^2-56x+16`

Algebraic Techniques, SMB-058

Expand and simplify the expression `(c-2)^2` (2 marks)

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`c^2-4c+4`

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`(c-2)^2`	`=(c-2)(c-2)`
	`=c^2-2c-2c+4`
	`=c^2-4c+4`

Algebraic Techniques, SMB-057

Expand and simplify the expression `(2x-5)^2` (2 marks)

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`4x^2-20x+25`

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`(2x-5)^2`	`=(2x-5)(2x-5)`
	`=4x^2-10x-10x+25`
	`=4x^2-20x+25`

Algebraic Techniques, SMB-056

Expand and simplify the expression `(2y-3)(2y+3)` (2 marks)

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`4y^2-9`

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`(2y-3)(2y+3)`	`=4y^2+6y-6y-9`
	`=4y^2-9`

Algebraic Techniques, SMB-055

Expand and simplify the expression `(p-4q)(p+4q)` (2 marks)

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`p^2-16q^2`

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`(p-4q)(p+4q)`	`=p^2+4pq-4pq-16q^2`
	`=p^2-16q^2`

Algebraic Techniques, SMB-054

Expand and simplify the expression `(3a+1)(2-a)+2a+4` (2 marks)

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`-3a^2+7a+6`

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`(3a+1)(2-a)+2a+4`	`=6a-3a^2+2-a+2a+4`
	`=-3a^2+7a+6`

Algebraic Techniques, SMB-048

Factorise the expression `b^2+4b-12` (2 marks)

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`(b+6)(b-2)`

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`b^2+4b-12=(b+6)(b-2)`

Algebraic Techniques, SMB-047

Factorise the expression `-a^2+3a+10` (2 marks)

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`-(a+2)(a-5)\ \ text(or)\ \ (5-a)(a+2)`

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`-a^2+3a+10`	`=-(a^2-3a-10)`
	`=-(a-5)(a+2)`
	`=(5-a)(a+2)`

Algebraic Techniques, SMB-046

Factorise the expression `n^2+5n-24` (2 marks)

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`(n+8)(n-3)`

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`n^2+5n-24=(n+8)(n-3)`

Algebraic Techniques, SMB-045

Fully factorise the expression `6x^2-8x-8` (2 marks)

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`2(3x+2)(x-2)`

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`6x^2-8x-8=2(3x+2)(x-2)`

Algebraic Techniques, SMB-044

Factorise the expression `2p^2-5p-12` (2 marks)

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`(2p+3)(p-4)`

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`2p^2-5p-12=(2p+3)(p-4)`

Algebraic Techniques, SMB-043

Factorise the expression `y^2-2y-15` (2 marks)

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`(y+3)(y-5)`

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`y^2-2y-15=(y+3)(y-5)`

Algebraic Techniques, SMB-042

Factorise the expression `x^2+6x+8` (2 marks)

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`(x+2)(x+4)`

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`x^2+6x+8=(x+2)(x+4)`

Algebraic Techniques, SMB-041

Expand and simplify the expression `(4x-3y)(y-x)` (2 marks)

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`-4x^2+7xy-3y^2`

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`(4x-3y)(y-x)`	`=4xy-4x^2-3y^2+3xy`
	`=-4x^2+7xy-3y^2`

Algebraic Techniques, SMB-040

Expand and simplify the expression `(7m-3)(m-2)` (2 marks)

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`7m^2-17m+6`

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`(7m-3)(m-2)`	`=7m^2-14m-3m+6`
	`=7m^2-17m+6`

Algebraic Techniques, SMB-039

Expand and simplify the expression `(a-b)(3a-2b)` (2 marks)

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`3a^2-5ab+2b^2`

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`(a-b)(3a-2b)`	`=3a^2-2ab-3ab+2b^2`
	`=3a^2-5ab+2b^2`

Algebraic Techniques, SMB-038

Expand and simplify the expression `(2p-3)(p+1)` (2 marks)

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`2p^2-p-3`

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`(2p-3)(p+1)`	`=2p^2+2p-3p-3`
	`=2p^2-p-3`

Algebraic Techniques, SMB-037

Expand and simplify the expression `(2x-1)(2x+1)` (2 marks)

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`4x^2-1`

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`(2x-1)(2x+1)`	`=4x^2+2x-2x-1`
	`=4x^2-1`

Algebraic Techniques, SMB-026

Expand and simplify the expression

`13-3(x-4)` (2 marks)

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`25-3x`

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`13-3(x-4)`	`=13-3x+12`
	`=25-3x`

Algebraic Techniques, SMB-025

Expand and simplify the expression

`3a(4a-5)-2(a-3)` (2 marks)

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`12a^2-17a+6`

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`3a(4a-5)-2(a-3)`	`=12a^2-15a-2a+6`
	`=12a^2-17a+6`

Algebraic Techniques, SMB-024

Expand the expression `-2(5x^2-3x-4)` (2 marks)

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`-10x^2+6x+8`

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`-2(5x^2-3x-4)=-10x^2+6x+8`

Algebraic Techniques, SMB-023

Expand the expression `3(2y^2-3y+1)` (2 marks)

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`6y^2-9y+3`

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`3(2y^2-3y+1)=6y^2-9y+3`

Algebraic Techniques, SMB-022

Expand and simplify the expression `4a(a-3)-5(6-a)` (2 marks)

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`4a^2-7a-30`

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`4a(a-3)-5(6-a)`	`=4a^2-12a-30+5a`
	`=4a^2-7a-30`

Algebraic Techniques, SMB-021

Expand and simplify the expression `3x(x-2)+4(x-5)` (2 marks)

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`3x^2-2x-20`

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`3x(x-2)+4(x-5)`	`=3x^2-6x+4x-20`
	`=3x^2-2x-20`

Algebraic Techniques, SMB-020 MC

Which of the following is always equal to `a-b`?

`-a-(-b)`
`-a+b`
`b-a`
`-b+a`

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`D`

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`a-b = -b + a`

`=>D`

Algebraic Techniques, SMB-019 MC

Which expression is equivalent to `4x^2-12x + x^3?`

`4(x^2-3x + x^3)`
`4x (x-3 + x^2)`
`x^2 (4-12 + x)`
`x (4x-12 + x^2)`

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`D`

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`x (4x-12 + x^2)`

`= 4x^2-12x + x^3`

`=>D`

Algebraic Techniques, SMB-018 MC

What expression is equivalent to `-(y-6)`?

`y-6`
`-y-6`
`y+6`
`-y+6`

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`D`

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`-(y-6) = -y-(-6)=-y + 6`

`=>D`

Algebraic Techniques, SMB-017 MC

Which expression is equal to `4x-8 + 3x + 2`?

`x`
`7x-6`
`7x-10`
`x-10`

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`B`

Show Worked Solution

`4x-8 + 3x + 2=7x-6`

`=>B`

Algebraic Techniques, SMB-016 MC

Which expression is equivalent to `12x + 24`?

`3(4x+8)`
`3(4x+21)`
`6(2x+18)`
`12(x+24)`

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`A`

Show Worked Solution

`3 (4x + 8)`	`= 3 xx 4x + 3 xx 8`
	`= 12x + 24`

`=>A`

Algebraic Techniques, SMB-015 MC

Which expression is equivalent to `5-6t`?

`6-5t`
`6t-5`
`-5+6t`
`-6t+5`

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`D`

Show Worked Solution

`-6t+5`

`=>D`

Algebraic Techniques, SMB-014 MC

Which one of the following expressions is equivalent to `4 (3m-1)`?

`7m-1`
`12m-1`
`12m-4`
`8m`

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`C`

Show Worked Solution

`4 (3m-1)`

`= (4 xx 3m)-(4 xx 1)`

`= 12m-4`

`=>C`

Algebraic Techniques, SMB-013 MC

The expression `3x + 7 + 8x + 11` can also be written as

`10x+19`
`11x + 18`
`18x+11`
`29x`

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`B`

Show Worked Solution

`3x + 7 + 8x + 11 = 11x + 18`

`=>B`

Algebra, STD2 A1 2016 HSC 5 MC

Which expression is equivalent to `2(3x-4) + 2`?

`6x-2`
`6x-4`
`6x-6`
`6x-10`

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`C`

Show Worked Solution

`2(3x-4) + 2`

`= 6x-8 + 2`

`= 6x-6`

`=> C`

Algebra, STD2 A1 2015 HSC 2 MC

Which of the following is `4x + 3y-x-5y` in its simplest form?

`3x - 2y`
`3x + 8y`
`5x - 2y`
`5x + 8y`

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`A`

Show Worked Solution

`4x + 3y-x-5y`

`= 3x-2y`

`⇒ A`

Functions, 2ADV F1 2006 HSC 1b

Factorise `2x^2 + 5x-3`. (2 marks)

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`(2x-1) (x + 3)`

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`2x^2 + 5x-3= (2x-1) (x + 3)`

Algebra, STD2 A1 2009 HSC 25a

Simplify `5-2(x + 7)`. (2 marks)

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`-2x-9`

Show Worked Solution

♦ Mean mark 47%

`5-2(x + 7)`	`= 5-2x-14`
	`= -2x-9`