Use synthetic division to decompose the following polynomial into two factors knowing that (x + 2) is a factor: 10x3+11x2-27x-18

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To decompose the polynomial 10x3+11x227x18 using synthetic division by the factor (x+2), we first set x+2=0, which gives us x=2.

Next, we can perform synthetic division using 2, and the coefficients of the polynomial 10,11,27,18:

  1. Write down the coefficients: 210112718  

  2. Bring down the leading coefficient (10): 210112718  10 

  3. Multiply 2 by 10 and add to the next coefficient (11): 2×10=20, and 11+(20)=9: 210112718 20 109 

  4. Multiply 2 by 9 and add to 27: 2×9=18, and 27+18=9: 210112718 2018 1099 

  5. Multiply 2 by 9 and add to 18: 2×9=18, and 18+18=0: 210112718 20180 10990 

The result of the synthetic division is: 10x29x9 Since the remainder is 0, we can confirm that x+2 is a factor of the original polynomial.

Next, we can factor the quadratic 10x29x9. To factor this quadratic, we can use the quadratic formula: x=b±b24ac2a where a=10, b=9, and c=9.

Calculating the discriminant: b24ac=(9)2410(9)=81+360=441

Now calculating the roots: x=9±44120=9±2120 Calculating the two roots:

  1. x=3020=32
  2. x=1220=35

Thus, we can factor the quadratic as: 10x29x9=10(x32)(x+35)

However, for simplicity, we can also write it as: 10(x32)(x+35)=(x+2)(10x29x9)

The complete factorization can be written as: 10(x+2)(x32)(x+35)

Summarizing, the factors of 10x3+11x227x18 are: (x+2)(10x29x9)