Synthetic Division to Find Additional Zeros
When one zero c of a polynomial P(x) is known, synthetic division can be used to divide P(x) by (x - c) to obtain P(x) = (x - c) \cdot Q(x), where Q(x) is a polynomial of lower degree whose zeros are the remaining zeros of P(x). Synthetic division is a streamlined method for dividing a polynomial by a linear factor (x - c) when c is a known zero. This process reduces the degree of the polynomial by one, making it easier to find the remaining zeros. This skill is part of Grade 11 math in enVision, Algebra 2.
Key Concepts
When one zero $c$ of a polynomial $P(x)$ is known, synthetic division can be used to divide $P(x)$ by $(x c)$ to obtain $P(x) = (x c) \cdot Q(x)$, where $Q(x)$ is a polynomial of lower degree whose zeros are the remaining zeros of $P(x)$.
Common Questions
What is Synthetic Division to Find Additional Zeros?
When one zero c of a polynomial P(x) is known, synthetic division can be used to divide P(x) by (x - c) to obtain P(x) = (x - c) \cdot Q(x), where Q(x) is a polynomial of lower degree whose zeros are the remaining zeros of P(x)..
How does Synthetic Division to Find Additional Zeros work?
Example: Given P(x) = x^3 - 6x^2 + 11x - 6 with known zero x = 1, use synthetic division with c = 1 to get P(x) = (x - 1)(x^2 - 5x + 6) = (x - 1)(x - 2)(x - 3), so all zeros are x = 1, 2, 3.
Give an example of Synthetic Division to Find Additional Zeros.
For P(x) = 2x^3 + x^2 - 13x + 6 with known zero x = 2, synthetic division gives P(x) = (x - 2)(2x^2 + 5x - 3) = (x - 2)(2x - 1)(x + 3), so zeros are x = 2, \frac{1}{2}, -3.
Why is Synthetic Division to Find Additional Zeros important in math?
Synthetic division is a streamlined method for dividing a polynomial by a linear factor (x - c) when c is a known zero. This process reduces the degree of the polynomial by one, making it easier to find the remaining zeros.
What grade level covers Synthetic Division to Find Additional Zeros?
Synthetic Division to Find Additional Zeros is a Grade 11 math topic covered in enVision, Algebra 2 in Chapter 3: Polynomial Functions. Students at this level study the concept as part of their grade-level standards and are expected to explain, analyze, and apply what they have learned.
What are typical Synthetic Division to Find Additional Zeros problems?
Given P(x) = x^3 - 6x^2 + 11x - 6 with known zero x = 1, use synthetic division with c = 1 to get P(x) = (x - 1)(x^2 - 5x + 6) = (x - 1)(x - 2)(x - 3), so all zeros are x = 1, 2, 3.; For P(x) = 2x^3 + x^2 - 13x + 6 with known zero x = 2, synthetic division gives P(x) = (x - 2)(2x^2 + 5x - 3) = (x - 2)(2x - 1)(x + 3), so zeros are x = 2, \frac{1}{2}, -3.; Given P(x) = x^4 - 5x^3 + 5x^2 + 5x - 6 with known zero x = 1, synthetic division yields P(x)