What occurs when the expression f(x) = 4x^3 + 3x^2 + ax + b is divided by x – 1 and x + 1?

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Understanding Polynomial Division

When you divide a polynomial by a linear expression, you can gain insights into the behavior of the polynomial, such as its roots and factors. In this case, we’re looking at the polynomial f(x) = 4x³ + 3x² + ax + b and dividing it by the factors x – 1 and x + 1.

1. Polynomial Division by x – 1

When we divide the polynomial by x – 1, we can use the Remainder Theorem. This theorem states that the remainder of the division of a polynomial f(x) by x – c is equal to f(c). Here, we need to calculate:

  • Let c = 1:
    • Substituting x = 1 into f(x):
      f(1) = 4(1)³ + 3(1)² + a(1) + b = 4 + 3 + a + b = 7 + a + b

This means that the remainder when f(x) is divided by x – 1 is 7 + a + b.

2. Polynomial Division by x + 1

Next, we apply the same principle for x + 1. Again applying the Remainder Theorem, we find:

  • Let c = -1:
    • Substituting x = -1 into f(x):
      f(-1) = 4(-1)³ + 3(-1)² + a(-1) + b = -4 + 3 – a + b = -1 – a + b

This means that the remainder when f(x) is divided by x + 1 is -1 – a + b.

3. Conclusion

To summarize, when dividing the polynomial f(x) = 4x³ + 3x² + ax + b by x – 1, the remainder is 7 + a + b, and when divided by x + 1, the remainder is -1 – a + b. These results not only give us the remainders but also indicate how the polynomial behaves at specific points. Understanding these divisions can be crucial for further analysis, like finding roots or simplifying the polynomial.

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