PM Review Submission History Week 8 Assignment Operation

12821 851 Pm Review Submission History Week 8 Assignment Operat

Form a polynomial whose real zeros and degree are given. Zeros: -44, 0, 11; degree: 3. Type a polynomial with integer coefficients and a leading coefficient of 1.

Form a polynomial whose zeros and degree are given. Zeros: -3 (multiplicity 1), -4 (multiplicity 2); degree 3. Type a polynomial with integer coefficients and a leading coefficient of 1.

For the polynomial function below: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of |x|.

f(x) = -7(x – 4)(x + 5)^2

For the polynomial function below: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the behavior of the graph near each x-intercept. (d) Determine the maximum number of turning points on the graph. (e) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of |x|.

f(x) = -6(x^2 + 4)(x – 7)^3

Paper For Above instruction

Constructing Polynomial Functions from Zeros

In algebra and calculus, the process of constructing polynomial functions from specified zeros involves translating the given roots into factors of the polynomial. When the zeros are real numbers, each zero corresponds to a factor of the form (x - zero). The multiplicity of a zero indicates how many times that factor appears. To form a polynomial with integer coefficients and a leading coefficient of 1, one multiplies out the factors, ensuring the polynomial’s leading term is monic.

For the first scenario, with zeros at -44, 0, and 11 and degree 3, the polynomial can be constructed directly as the product of factors derived from these zeros. Since the zeros are all real and the polynomial degree is 3, each zero is associated with a factor, and the polynomial takes the form:

f(x) = (x + 44) · x · (x - 11)

This polynomial is already monic, with leading coefficient 1, and expanded form provides an explicit expression:

f(x) = (x + 44)(x)(x - 11) = x(x + 44)(x - 11)

Expanding step-by-step:

• First, expand (x + 44)(x - 11):
• (x + 44)(x - 11) = x^2 - 11x + 44x - 484 = x^2 + 33x - 484
• Now, multiply by x:
• f(x) = x(x^2 + 33x - 484) = x^3 + 33x^2 - 484x

Thus, the polynomial is:

f(x) = x^3 + 33x^2 - 484x

For the second scenario, with zeros at -3 (multiplicity 1) and -4 (multiplicity 2), the zeros translate into factors:

• Zero at -3: (x + 3)
• Zero at -4 with multiplicity 2: (x + 4)^2

Combining these factors yields:

f(x) = (x + 3)(x + 4)^2

This polynomial has a leading coefficient of 1 and is already factored. To express it in expanded form, perform polynomial multiplication:

• Expand (x + 4)^2: (x + 4)^2 = x^2 + 8x + 16
• Multiply by (x + 3):

f(x) = (x + 3)(x^2 + 8x + 16)

Expanding:

• x · (x^2 + 8x + 16) = x^3 + 8x^2 + 16x
• 3 · (x^2 + 8x + 16) = 3x^2 + 24x + 48
• Adding these, the polynomial becomes:

f(x) = x^3 + (8x^2 + 3x^2) + (16x + 24x) + 48 = x^3 + 11x^2 + 40x + 48

The polynomial with integer coefficients and monic leading term is therefore f(x) = x³ + 11x² + 40x + 48.

Analyzing Polynomial Graph Behavior

Given the polynomial functions, analyzing their roots, multiplicities, and behaviors at intercepts helps understand their graphs.

Function: f(x) = -7(x - 4)(x + 5)^2

(a) List each real zero and its multiplicity:

• Zero at x = 4, multiplicity 1
• Zero at x = -5, multiplicity 2

(b) Determine whether the graph crosses or touches the x-axis at each intercept:

• At x=4: multiplicity 1 (odd), so the graph crosses the x-axis.
• At x=-5: multiplicity 2 (even), so the graph touches and tangents the x-axis without crossing.

(c) Behavior near each x-intercept:

• x=4: the graph crosses the x-axis and changes sign.
• x=-5: the graph touches the x-axis and turns around, showing a local extremum.

(d) Maximum number of turning points: For degree 3, maximum possible turning points is 2.

(e) End behavior: As x → ±∞, the dominant term is -7x² · x (from expansion). Since the degree is 3 (odd) with a negative leading coefficient, as x → ∞, f(x) → -∞, and as x → -∞, f(x) → ∞. The graph resembles the end behavior of a cubic function with a negative leading coefficient, tending downward as x becomes large positive and upward as x becomes large negative.

Function: f(x) = -6(x^2 + 4)(x – 7)^3

(a) List each real zero and its multiplicity:

• x=0, zero of (x^2 + 4): no real zeros from x^2 + 4 (since x^2 + 4 ≠ 0 for real x), so only zero at x=7, multiplicity 3.
• Zero at x=7, multiplicity 3.

(b) Cross or touch at intercepts:

• x=7: multiplicity 3 (odd), so the graph crosses the x-axis.

(c) Behavior near x=7:

• Graph crosses x-axis with a sign change due to odd multiplicity.

(d) Maximum number of turning points: a degree 5 polynomial (since degree is Get decrypted by expanding the factors) can have up to 4 turning points.

(e) End behavior: The polynomial is degree 5 with leading coefficient -6. Since the degree is odd and the leading coefficient negative, as x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞. The graph exhibits typical cubic-like end behavior with downward trend for large positive x values and upward trend for negative x values.

Conclusion

Constructing and analyzing polynomial functions based on their zeros and multiplicities is foundational in understanding their graphing and behavior. The methods involve transforming zeros into factors, expanding when necessary, and studying the impacts on graph shape and end behavior. Recognizing the multiplicity and degree’s influence on crossing, touching, and the number of turning points enables deeper insights into polynomial function analysis, critical in advanced mathematics and calculus applications.

References

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