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(x − 2) Is A Factor Of X4 + 2x3 − 7x2 − 8x + 12. The Other Factors Are And

5 min read Jun 16, 2024
(x − 2) Is A Factor Of X4 + 2x3 − 7x2 − 8x + 12. The Other Factors Are And

Factoring a Polynomial

We are given that (x - 2) is a factor of the polynomial x⁴ + 2x³ − 7x² − 8x + 12. To find the other factors, we can use polynomial long division or synthetic division.

Using Polynomial Long Division

  1. Set up the division:

         ___________
x - 2 | x⁴ + 2x³ − 7x² − 8x + 12

  2. Divide the leading terms:

    • x⁴ divided by x is x³. Write x³ above the line.

    • Multiply (x - 2) by x³: x⁴ - 2x³

    • Subtract this result from the dividend:

         x³ ___________

    x - 2 | x⁴ + 2x³ − 7x² − 8x + 12 -(x⁴ - 2x³) ----------- 0 + 4x³

  3. Bring down the next term:

         x³ ___________
x - 2 | x⁴ + 2x³ − 7x² − 8x + 12 
         -(x⁴ - 2x³)
         -----------
               0 + 4x³ - 7x²

  4. Repeat the process:

    • 4x³ divided by x is 4x². Write 4x² above the line.

    • Multiply (x - 2) by 4x²: 4x³ - 8x²

    • Subtract:

         x³ + 4x² ___________

    x - 2 | x⁴ + 2x³ − 7x² − 8x + 12 -(x⁴ - 2x³) ----------- 0 + 4x³ - 7x² -(4x³ - 8x²) ----------- x²

  5. Continue bringing down terms and dividing:

         x³ + 4x² + x ___________
x - 2 | x⁴ + 2x³ − 7x² − 8x + 12 
         -(x⁴ - 2x³)
         -----------
               0 + 4x³ - 7x² 
               -(4x³ - 8x²)
               -----------
                        x² - 8x
                        -(x² - 2x)
                        -----------
                               -6x

  6. Bring down the last term and divide:

         x³ + 4x² + x - 6 ___________
x - 2 | x⁴ + 2x³ − 7x² − 8x + 12 
         -(x⁴ - 2x³)
         -----------
               0 + 4x³ - 7x² 
               -(4x³ - 8x²)
               -----------
                        x² - 8x
                        -(x² - 2x)
                        -----------
                               -6x + 12
                               -(-6x + 12)
                               -----------
                                     0

Therefore, we have:

x⁴ + 2x³ − 7x² − 8x + 12 = (x - 2)(x³ + 4x² + x - 6)

Finding the Other Factors

Now we need to factor the cubic polynomial (x³ + 4x² + x - 6). This can be a bit more challenging, and we can use techniques like the Rational Root Theorem or factoring by grouping. In this case, we can try factoring by grouping:

  1. Group the terms:

    (x³ + 4x²) + (x - 6)

  2. Factor out common factors:

    x²(x + 4) + 1(x - 6)

Unfortunately, we cannot factor further with this method. We can use the Rational Root Theorem to test for possible rational roots of the cubic polynomial. This theorem suggests that if there are any rational roots, they must be factors of the constant term (-6) divided by factors of the leading coefficient (1).

  • Factors of -6: ±1, ±2, ±3, ±6
  • Factors of 1: ±1

Therefore, possible rational roots are: ±1, ±2, ±3, ±6. By testing these values, we find that x = 1 is a root. This means (x - 1) is a factor of the cubic polynomial.

Using polynomial long division or synthetic division again, we can divide (x³ + 4x² + x - 6) by (x - 1) to get the remaining quadratic factor:

(x³ + 4x² + x - 6) / (x - 1) = x² + 5x + 6

This quadratic factors easily:

x² + 5x + 6 = (x + 2)(x + 3)

Conclusion

Therefore, the complete factorization of x⁴ + 2x³ − 7x² − 8x + 12 is:

(x - 2)(x - 1)(x + 2)(x + 3)

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