Math

Question7) Expand the expression (2x3)(x4)(2 x-3)(x-4). 8) Expand the expression (x5)(3x+4)(x-5)(3 x+4). 9) Expand the expression (x5)(x3)(x-5)(x-3). 10) Expand the expression (3x+8)(3x8)(3 x+8)(3 x-8). 11) Expand the expression (3x8)(3x4)(3 x-8)(3 x-4). 12) Simplify the expression 3x2+3x22x33 x^{2}+3 x^{2}-2 x^{3}.

Studdy Solution

STEP 1

Assumptions1. We are given six expressions to simplify. . We will use the distributive property (also known as the FOIL method for binomials) to simplify the expressions.
3. We will combine like terms where possible.

STEP 2

Let's start with the first expression (2x)(x4)(2x-)(x-4). We apply the distributive property.
(2x)(x4)=2xx+2x4+x+4(2x-)(x-4) =2x \cdot x +2x \cdot -4 + - \cdot x + - \cdot -4

STEP 3

implify the expression.
(2x3)(x)=2x28x3x+12(2x-3)(x-) =2x^2 -8x -3x +12

STEP 4

Combine like terms.
(2x3)(x4)=2x211x+12(2x-3)(x-4) =2x^2 -11x +12

STEP 5

Now, let's move to the second expression (x5)(3x+4)(x-5)(3x+4). We apply the distributive property.
(x5)(3x+4)=x3x+x4+53x+54(x-5)(3x+4) = x \cdot3x + x \cdot4 + -5 \cdot3x + -5 \cdot4

STEP 6

implify the expression.
(x5)(3x+4)=3x2+4x15x20(x-5)(3x+4) =3x^2 +4x -15x -20

STEP 7

Combine like terms.
(x5)(3x+4)=3x211x20(x-5)(3x+4) =3x^2 -11x -20

STEP 8

Next, let's simplify the third expression (x5)(x3)(x-5)(x-3). We apply the distributive property.
(x5)(x3)=xx+x3+5x+53(x-5)(x-3) = x \cdot x + x \cdot -3 + -5 \cdot x + -5 \cdot -3

STEP 9

implify the expression.
(x5)(x3)=x23x5x+15(x-5)(x-3) = x^2 -3x -5x +15

STEP 10

Combine like terms.
(x5)(x3)=x28x+15(x-5)(x-3) = x^2 -8x +15

STEP 11

Now, let's simplify the fourth expression (3x+8)(3x8)(3x+8)(3x-8). We apply the distributive property.
(3x+8)(3x8)=3x3x+3x8+83x+88(3x+8)(3x-8) =3x \cdot3x +3x \cdot -8 +8 \cdot3x +8 \cdot -8

STEP 12

implify the expression.
(x+8)(x8)=9x224x+24x64(x+8)(x-8) =9x^2 -24x +24x -64

STEP 13

Combine like terms.
(3x+8)(3x8)=9x264(3x+8)(3x-8) =9x^2 -64

STEP 14

Next, let's simplify the fifth expression (3x8)(3x4)(3x-8)(3x-4). We apply the distributive property.
(3x8)(3x4)=3x3x+3x4+83x+84(3x-8)(3x-4) =3x \cdot3x +3x \cdot -4 + -8 \cdot3x + -8 \cdot -4

STEP 15

implify the expression.
(3x8)(3x4)=9x212x24x+32(3x-8)(3x-4) =9x^2 -12x -24x +32

STEP 16

Combine like terms.
(3x8)(3x4)=9x236x+32(3x-8)(3x-4) =9x^2 -36x +32

STEP 17

Finally, let's simplify the sixth expression 3x2+3x22x33x^2+3x^2-2x^3. We combine like terms.
3x2+3x22x3=2x3+6x23x^2+3x^2-2x^3 = -2x^3 +6x^2

STEP 18

The simplified expressions are7) (2x3)(x4)=2x211x+12(2x-3)(x-4) =2x^2 -11x +12 8) (x5)(3x+4)=3x211x20(x-5)(3x+4) =3x^2 -11x -20 ) (x5)(x3)=x28x+15(x-5)(x-3) = x^2 -8x +15 10) (3x+8)(3x8)=x264(3x+8)(3x-8) =x^2 -64 11) (3x8)(3x4)=x236x+32(3x-8)(3x-4) =x^2 -36x +32 12) 3x2+3x22x3=2x3+6x23x^2+3x^2-2x^3 = -2x^3 +6x^2

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