I need help on this question? ## Polynomial Function AnalysisA polynomial function $ P $ and its graph are given:\[ P(x) = 2x^4 - 6x^3 + 2x^2 + 6x - 4 \]### Graph Description:The graph of the polynomial function is a smooth curve that intersects the y-axis at $-4$. The curve extends from $-2$ to $3$ on the x-axis and from $-6$ to $2$ on the y-axis. Key features of the graph include:- Peaks and valleys that suggest turning points in the polynomial.- The graph passes through the x-axis at specific points, indicating potential zeros.### Tasks:#### (a) List all possible rational zeros of $ P $ given by the Rational Zeros Theorem.- $ x = -1, 1, 0.5, -0.5, -2, 2, -4, 4 $- These values are presented as potential rational zeros based on the theorem.#### (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros.- $ x = 1, -1, 2 $- The provided answer appears to be incorrect according to the check mark.### Educational Insight:The Rational Zeros Theorem is a useful tool for identifying potential zeros of a polynomial function by applying mathematical reasoning based on the leading coefficient and constant term. Comparing the theoretical list of rational zeros with the graphically observed zeros provides an opportunity to explore the relationship between algebraic expressions and their visual representations.

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A polynomial function P and its graph are given. P(x) = 2x4 - 6x³ + 2x² + 6x-4 -2 y 2 6 X (a) List all possible rational zeros of P given by the Rational Zeros Theorem. (Enter your answers as a comma-separated list.) x= -1,1,0.5, -0.5, -2,2,- 4,4 W ✓ (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.) x = 1,-1,2

A polynomial function P and its graph are given. P(x) = 2x4 - 6x³ + 2x² + 6x-4 -2 y 2 6 X (a) List all possible rational zeros of P given by the Rational Zeros Theorem. (Enter your answers as a comma-separated list.) x= -1,1,0.5, -0.5, -2,2,- 4,4 W ✓ (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.) x = 1,-1,2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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I need help on this question?

## Polynomial Function Analysis

A polynomial function $ P $ and its graph are given:

\[ P(x) = 2x^4 - 6x^3 + 2x^2 + 6x - 4 \]

### Graph Description:

The graph of the polynomial function is a smooth curve that intersects the y-axis at $-4$. The curve extends from $-2$ to $3$ on the x-axis and from $-6$ to $2$ on the y-axis. Key features of the graph include:

- Peaks and valleys that suggest turning points in the polynomial.
- The graph passes through the x-axis at specific points, indicating potential zeros.

### Tasks:

#### (a) List all possible rational zeros of $ P $ given by the Rational Zeros Theorem.

- $ x = -1, 1, 0.5, -0.5, -2, 2, -4, 4 $
- These values are presented as potential rational zeros based on the theorem.

#### (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros.

- $ x = 1, -1, 2 $
- The provided answer appears to be incorrect according to the check mark.

### Educational Insight:

The Rational Zeros Theorem is a useful tool for identifying potential zeros of a polynomial function by applying mathematical reasoning based on the leading coefficient and constant term. Comparing the theoretical list of rational zeros with the graphically observed zeros provides an opportunity to explore the relationship between algebraic expressions and their visual representations.

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