Math Problem Statement

Is x2+3x1 a factor of x3+2x25x6x^2+3x-1\text{ a factor of }x^3+2x^2-5x-6

Solution

To determine if x2+3x1x^2 + 3x - 1 is a factor of x3+2x25x6x^3 + 2x^2 - 5x - 6, we can perform polynomial division or use synthetic division.

Step 1: Set up the division

We will divide x3+2x25x6x^3 + 2x^2 - 5x - 6 by x2+3x1x^2 + 3x - 1.

Step 2: Perform the division

  1. Divide the leading term of the dividend x3x^3 by the leading term of the divisor x2x^2: x3x2=x\frac{x^3}{x^2} = x So, the first term of the quotient is xx.

  2. Multiply the entire divisor x2+3x1x^2 + 3x - 1 by xx and subtract it from the original polynomial: (x2+3x1)×x=x3+3x2x(x^2 + 3x - 1) \times x = x^3 + 3x^2 - x Subtract this from the original polynomial: (x3+2x25x6)(x3+3x2x)=x24x6(x^3 + 2x^2 - 5x - 6) - (x^3 + 3x^2 - x) = -x^2 - 4x - 6

  3. Repeat the process with the new polynomial x24x6-x^2 - 4x - 6: Divide the leading term x2-x^2 by the leading term x2x^2: x2x2=1\frac{-x^2}{x^2} = -1 So, the next term in the quotient is 1-1.

  4. Multiply the entire divisor x2+3x1x^2 + 3x - 1 by 1-1 and subtract: (x2+3x1)×(1)=x23x+1(x^2 + 3x - 1) \times (-1) = -x^2 - 3x + 1 Subtract this from x24x6-x^2 - 4x - 6: (x24x6)(x23x+1)=x7(-x^2 - 4x - 6) - (-x^2 - 3x + 1) = -x - 7

Step 3: Conclusion

Since the remainder is x7-x - 7, which is not zero, x2+3x1x^2 + 3x - 1 is not a factor of x3+2x25x6x^3 + 2x^2 - 5x - 6.

Would you like to see a detailed explanation or have any questions about this process?

Here are five related questions for further exploration:

  1. How can you use the Remainder Theorem to check if a polynomial is a factor?
  2. What is synthetic division, and how does it differ from polynomial division?
  3. How would you factor x3+2x25x6x^3 + 2x^2 - 5x - 6 completely?
  4. What is the relationship between roots of a polynomial and its factors?
  5. Can x2+3x1x^2 + 3x - 1 divide another polynomial, like x3+5x2+2x1x^3 + 5x^2 + 2x - 1?

Tip: Always verify your results by multiplying the quotient and the divisor to ensure the division was performed correctly.

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Math Problem Analysis

Mathematical Concepts

Polynomial division
Synthetic division

Formulas

-

Theorems

Remainder Theorem

Suitable Grade Level

Advanced High School