Math  /  Algebra

QuestionAssignment: Show complete solution 1 Perform the following multiplications and simplify your results: (a) (8x4)(4x23x+2)(8 x-4)\left(4 x^{2}-3 x+2\right) (b) (2x+3)(5x3+3x4)(2 x+3)\left(5 x^{3}+3 x-4\right)

Studdy Solution

STEP 1

1. The given expressions involve polynomial multiplication, which requires distributing each term in the first polynomial to each term in the second polynomial.
2. Simplifying the results involves combining like terms.
3. The final expressions should be simplified fully.

STEP 2

1. Multiply and simplify the expression (8x4)(4x23x+2)(8x-4)(4x^2-3x+2).
2. Multiply and simplify the expression (2x+3)(5x3+3x4)(2x+3)(5x^3+3x-4).

STEP 3

Distribute each term in (8x4)(8x-4) to each term in (4x23x+2)(4x^2-3x+2).
(8x4)(4x23x+2)=8x(4x2)+8x(3x)+8x(2)4(4x2)4(3x)4(2)(8x - 4)(4x^2 - 3x + 2) = 8x(4x^2) + 8x(-3x) + 8x(2) - 4(4x^2) - 4(-3x) - 4(2)

STEP 4

Multiply the terms and simplify.
8x4x2=32x38x \cdot 4x^2 = 32x^3 8x(3x)=24x28x \cdot (-3x) = -24x^2 8x2=16x8x \cdot 2 = 16x 44x2=16x2-4 \cdot 4x^2 = -16x^2 4(3x)=12x-4 \cdot (-3x) = 12x 42=8-4 \cdot 2 = -8
(8x4)(4x23x+2)=32x324x2+16x16x2+12x8(8x - 4)(4x^2 - 3x + 2) = 32x^3 - 24x^2 + 16x - 16x^2 + 12x - 8

STEP 5

Combine like terms.
32x3+(24x216x2)+(16x+12x)8=32x340x2+28x832x^3 + (-24x^2 - 16x^2) + (16x + 12x) - 8 = 32x^3 - 40x^2 + 28x - 8
(8x4)(4x23x+2)=32x340x2+28x8(8x - 4)(4x^2 - 3x + 2) = 32x^3 - 40x^2 + 28x - 8

STEP 6

Distribute each term in (2x+3)(2x+3) to each term in (5x3+3x4)(5x^3+3x-4).
(2x+3)(5x3+3x4)=2x(5x3)+2x(3x)+2x(4)+3(5x3)+3(3x)+3(4)(2x + 3)(5x^3 + 3x - 4) = 2x(5x^3) + 2x(3x) + 2x(-4) + 3(5x^3) + 3(3x) + 3(-4)

STEP 7

Multiply the terms and simplify.
2x5x3=10x42x \cdot 5x^3 = 10x^4 2x3x=6x22x \cdot 3x = 6x^2 2x4=8x2x \cdot -4 = -8x 35x3=15x33 \cdot 5x^3 = 15x^3 33x=9x3 \cdot 3x = 9x 34=123 \cdot -4 = -12
(2x+3)(5x3+3x4)=10x4+6x28x+15x3+9x12(2x + 3)(5x^3 + 3x - 4) = 10x^4 + 6x^2 - 8x + 15x^3 + 9x - 12

STEP 8

Combine like terms.
10x4+15x3+6x2+(8x+9x)12=10x4+15x3+6x2+x1210x^4 + 15x^3 + 6x^2 + (-8x + 9x) - 12 = 10x^4 + 15x^3 + 6x^2 + x - 12
(2x+3)(5x3+3x4)=10x4+15x3+6x2+x12(2x + 3)(5x^3 + 3x - 4) = 10x^4 + 15x^3 + 6x^2 + x - 12
Solution to (a): (8x4)(4x23x+2)=32x340x2+28x8(8x - 4)(4x^2 - 3x + 2) = 32x^3 - 40x^2 + 28x - 8
Solution to (b): (2x+3)(5x3+3x4)=10x4+15x3+6x2+x12(2x + 3)(5x^3 + 3x - 4) = 10x^4 + 15x^3 + 6x^2 + x - 12

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