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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+11x2−27x−18 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: −21011−27−18  

  2. Bring down the leading coefficient (10): −21011−27−18  10 

  3. Multiply −2 by 10 and add to the next coefficient (11): −2×10=−20, and 11+(−20)=−9: −21011−27−18 −20 10−9 

  4. Multiply −2 by −9 and add to −27: −2×−9=18, and −27+18=−9: −21011−27−18 −2018 10−9−9 

  5. Multiply −2 by −9 and add to −18: −2×−9=18, and −18+18=0: −21011−27−18 −20180 10−9−90 

The result of the synthetic division is: 10x2−9x−9 Since the remainder is 0, we can confirm that x+2 is a factor of the original polynomial.

Next, we can factor the quadratic 10x2−9x−9. To factor this quadratic, we can use the quadratic formula: x=−b±b2−4ac2a where a=10, b=−9, and c=−9.

Calculating the discriminant: b2−4ac=(−9)2−4⋅10⋅(−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: 10x2−9x−9=10(x−32)(x+35)

However, for simplicity, we can also write it as: 10(x−32)(x+35)=(x+2)(10x2−9x−9)

The complete factorization can be written as: 10(x+2)(x−32)(x+35)

Summarizing, the factors of 10x3+11x2−27x−18 are: (x+2)(10x2−9x−9)