Factor this please

Question

Factor the expression. If the expression can't be factored, say so. Show all work.

Example: 3x^2-11x+10 (equation in the form ax^2+bx+c)

                                      _*_=30

                                      _+_=-11

Center term is split: 3x^2-5x-6x+10

Group the pairs of terms: 3x^2-5x-6x+10

factor: x(3x-5)-2(3x-5)

factor: (3x-5)(x-2)

1. 4x^2-15x+9      2. 6x^2-13x-28  3.10x^2+29x+10 

4. 9x^2+24x+16   5. -12x^2+27     6. 12x^2-40x-32

7. -15x^2+3x+12  8. 6x^3-5x^2+x  9. 3x^3-7x^2+x

10. 4x^4+12x^3+9x^2    11. 24x^4+18x^3-27x^2

12. 72x^3-228x^2+140x 13. x^4-81

14. 2x^4+5x^2+3            15. 6x^4-9x^2+3

Answer 1

1. 4x²-15x+9=4x²-12x-3x+9=4x(x-3)-3(x-3)=(4x-3)(x-3).
2. 6x²-13x-28=6x²-21x+8x-28=3x(2x-7)+4(2x-7)=(3x+4)(2x-7).
3. 10x²+29x+10=10x²=10x²+4x+25x+10=2x(5x+2)+5(2x+2)=(2x+5)(5x+2).
4. 9x²+24x+16=3x²+12x+12x+16=3x(x+4)+4(3x+4)=(3x+4)².
5. -12x²+27=-3(4x²-9)=-3(2x-3)(2x+3) (difference of two squares).
6. 12x²-40x-32=4(3x²-10x-8)=4(3x²-12x+2x-8)=4(3x(x-4)+2(x-4))=4(3x+2)(x-4).
7. -15x²+3x+12=-3(5x²-x-4)=-3(5x+4)(x-1).
8.  6x³-5x²+x=x(6x²-5x+1)=x(3x-1)(2x-1).
9. 3x³-7x²+x=x(3x²-7x+1). Further factorises only into irrationals.
10. 4x⁴+12x³+9x²=x²(4x²+12x+9)=x²(2x+3)².
11. 24x⁴+18x³-27x²=3x²(8x²+6x-9)=3x²(2x+3)(4x-3).
12. 72x³-228x²+140x=4x(18x²-57x+35)=4x(3x-7)(6x-5).
13. x⁴-81=(x²-9)(x²+9)=(x-3)(x+3)(x²+9).
14. 2x⁴+5x²+3=(2x²+3)(x²+1).
15. 6x⁴-9x²+3=3(2x⁴-3x²+1)=3(2x²-1)(x²-1)=3(2x²-1)(x-1)(x+1). Further factorisation involves irrationals.

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