6.2 - Polynomial Divison

Polynomials Higher Degree and Root-Finding

Overview

• Focus on finding zeros/roots of polynomials, specifically for larger degrees.
• Goals
  • Understanding polynomial division:
  • Polynomial long division
  • Finding x-intercepts

Long Division Review

• Demonstration of long division using integers:
  • Example: Divide 7645 by 21.
  • Calculation steps:
  • 21 into 76: goes 3 times. (3 × 21 = 63) Remainder: 76 - 63 = 13.
  • Bring down 4: 21 into 134: goes 6 times. (6 × 21 = 126) Remainder: 134 - 126 = 8.
  • Bring down 5: 21 into 85: goes 4 times. (4 × 21 = 84) Remainder: 85 - 84 = 1.
  • Final representation: 364 + 1/21 or as a fraction:
    $7645 = 364 imes 21 + 1$

Polynomial Long Division

• Steps for polynomial long division:
  1. Arrange terms in descending order of powers.
  2. Divide the leading term of the dividend by the leading term of the divisor, record as first term of quotient.
  3. Multiply the whole divisor by the first term and align under the dividend.
  4. Subtract, change signs, and bring down the next term.
  5. Repeat until degree of remainder is less than degree of divisor.
• Example: Divide $x^2 + 2x - 24$ by $x + 6$:
  • First term: $x^2 / x = x$
  • Multiply: $x(x + 6) = x^2 + 6x$
  • Subtract: (2x - 6x) gives -4x, bring down -24.
  • Next: $-4x / x = -4$; Multiply: $-4(x + 6) = -4x - 24$
  • Subtract to find the remainder: 0.
• Conclusion: $x + 6$ and $x - 4$ are factors; polynomial can be represented as $x^2 + 2x - 24 = (x + 6)(x - 4)$

Division Algorithm Definition

• Given polynomials $p(x)$ and $d(x)$ (where $d(x) <br />\neq 0$), there exist unique polynomials $q(x)$ and $r(x)$:
  $p(x) = d(x) imes q(x) + r(x)$
  where the degree of $r(x)$ is less than the degree of $d(x)$.

More Complex Example

• Divide a polynomial with higher degree: (e.g., $6x^5 - 2x^4 + 9x^3 - 15x^2 - 19$ by $2x^2 - 3x + 1$).
• Take each term, perform polynomial long division systematically.

Complex Number Division

• Example: Divide by $x^2 + i$ where a polynomial has zeros in complex numbers (guidelines and placeholder strategy).
• Introduce placeholders for coefficients in polynomials where needed.

Finding Roots Using Polynomial Division

• Important to find polynomial factors which help determine x-intercepts (roots).
• Example: Using factor $x - c$ helps to find where $p(c) = 0$ ensures c is a zero of the polynomial polynomial:
  • Direct connection to graphing x-intercepts.
• Factoring larger polynomials into simpler factors, perform polynomial long division iteratively to find remaining factors.

Key Theorems

Remainder Theorem

• States that if a polynomial $p(x)$ is divided by $x - c$, the remainder equals $p(c)$.
• Method exemplifies how remainder enables finding roots or determining if a number is a root.

Synthetic Division

• Useful for quickly determining whether a number is a root of polynomial.
• Works specifically when dividing by a linear binomial $x - c$.
• Steps:
  1. Write down coefficients of the polynomial.
  2. Evaluate using the value of c in synthetic manner (multiply and add sequentially).
• Example:
  • Determine if 5 is a root of $-2x^4 + 11x^3 - 5x + 15$.

Constructing Polynomials from Given Roots

• Given roots, can reconstruct a polynomial either directly or through synthetic division for verification,
• Example of factors leads to constructing full polynomial through multiplication of roots demonstrated step-wise.

Y-Intercept and Adjustments

• When constructing, adjustments are needed based on specific y-intercept conditions (scaling the polynomial). Example found a specific point through substitution and multiplying appropriately at the end.

Additional Notes

• Ensure to maintain rigor in every step, particularly with changing signs and distributing dynamically through polynomial long division.
• Foster practice with both methods (long division and synthetic) to increase comfort with polynomial manipulations and understanding.

Conclusion

Understanding polynomial division, whether through traditional long division or synthetic division, is essential in finding roots and simplifying polynomial expressions effectively to draw insights about their behaviors (such as intercepts).