Perform A Series Of Connected Algebraic Equal Successions

Perform A Series Of Connected Algebraic Equal Successions To Sim

Perform a series of connected, algebraic, equal successions to simplify each complex number expression a) (3ð‘–−2)(4-5 ð‘–) b) ð‘– + 8 
 c) ð‘–^- ð‘– 2. Given the quadratic function f(x)=-(x-3) ^2 +16. Perform the following: a) Write the vertex of the parabola. Expand the equation to express in general form. 
 b) Find the y-intercept and the x-intercepts. Show all work. Make a table of vertex points. 
 c) Plot and label the vertex, y-intercept, x-intercepts and the axis of symmetry on the x-y plane. 
 d) Use vertex points to find additional points. Label point symmetric to y-intercept. 
 e) Connect points to form parabola. Write equation on the graph. 
 5. Given vertex = (−3, 8) and y-intercept = (0, -10) of a parabolic function, perform the following: a) Use given points to find d(x)= the equation of the parabola in vertex form. b) Find the x-intercepts. Make a table of vertex points. How does the parabola open and why? c) Plot and label the vertex, y-intercept, x-intercepts and the axis of symmetry on the x-y plane. d) Use vertex points to find additional points. Label point symmetric to y-intercept. e) Connect points to form parabola. Write equation on the graph. 6. Given the following p(x)= x^4 + 2x^3 -7x^2 – 8x +12 and p(1)=0, perform the following a) Use polynomial long division to factor (x). Continue to factor the quotient to linear factors. b) Write p(x) = in its complete factored form. c) Show that the given zero follows from your factored p(x) and find the three additional zeros.

Paper For Above instruction

This comprehensive mathematical analysis involves a sequence of algebraic manipulations, quadratic and polynomial functions, and graphing techniques. The tasks encompass simplifying complex numbers, analyzing quadratic functions, and factoring higher-degree polynomials, which are fundamental skills in algebra and precalculus studies.

Simplification of Complex Number Expressions

The first set of problems requires simplifying complex algebraic expressions involving complex numbers and exponents. For instance, expressions like (3(ỳ) – 2)(4 - 5(ỳ)) and (ỳ)^- (ỳ) – need to be carefully expanded using distributive laws and rules of exponents. When simplifying (3(ỳ) – 2)(4 - 5(ỳ)), distributive property yields:

(3(ỳ) – 2)(4 - 5(ỳ)) = 3(ỳ) 4 + 3(ỳ) (-5(ỳ)) - 2 4 - 2 (-5(ỳ)) = 12(ỳ) - 15(ỳ)^2 - 8 + 10(ỳ).

Similarly, simplifying (ỳ)^- (ỳ) involves understanding negative exponents, which convert to reciprocal powers. For example, (ỳ)^- (ỳ) = 1 / (ỳ), if interpreted as reciprocal. Ensuring proper handling of conjugates or imaginary components (if applicable) is essential in complex algebra.

Analysis and Graph of Quadratic Functions

Next, the quadratic function f(x) = -(x - 3)^2 + 16 is analyzed thoroughly. The vertex form indicates a parabola opening downward with vertex at (3, 16). To express in general form, expansion gives:

f(x) = - (x^2 - 6x + 9) + 16 = -x^2 + 6x - 9 + 16 = -x^2 + 6x + 7.

The y-intercept is found by evaluating f(0): y = -0 + 0 + 7 = 7. The x-intercepts are obtained by solving:

0 = -x^2 + 6x + 7, or equivalently, x^2 - 6x - 7 = 0. Using the quadratic formula yields:

x = [6 ± √(36 + 28)] / 2 = [6 ± √64] / 2 = [6 ± 8] / 2.

Thus, x = (6 + 8)/2 = 14/2 = 7, and x = (6 - 8)/2 = -2/2 = -1.

A table of key points includes the vertex (3, 16), y-intercept (0, 7), and x-intercepts (-1, 0) and (7, 0). Graphing these points with the axis of symmetry at x=3 provides a complete parabola.

Constructing a Parabola With Given Vertex and Y-intercept

Given the vertex (-3, 8) and y-intercept (0, -10), the quadratic function is in vertex form:

d(x) = a(x + 3)^2 + 8.

Using the y-intercept to find 'a':

-10 = a(0 + 3)^2 + 8 ⇒ -10 = 9a + 8 ⇒ 9a = -18 ⇒ a = -2.

Therefore, the equation is d(x) = -2(x + 3)^2 + 8.

The x-intercepts are found by setting d(x) = 0:

0 = -2(x + 3)^2 + 8 ⇒ -2(x + 3)^2 = -8 ⇒ (x + 3)^2 = 4 ⇒ x + 3 = ±2,

which gives x = -1 and x = -5.

Plotting the vertex at (-3,8), y-intercept at (0, -10), and x-intercepts at -1 and -5 illustrates the parabola opening downward (since a = -2), consistent with the negative coefficient.

Factoring a Quartic Polynomial and Finding Its Zeros

The polynomial p(x) = x^4 + 2x^3 - 7x^2 - 8x + 12 has a known root at x = 1. Polynomial division by (x - 1) proceeds as follows:

Set up to divide p(x) by (x - 1):

Using synthetic division:

• Coefficients: 1 | 2 | -7 | -8 | 12
• Bring down 1.
• Multiply 1 * 1 = 1, add to 2: 2 + 1 = 3.
• Multiply 3 * 1 = 3, add to -7: -7 + 3 = -4.
• Multiply -4 * 1 = -4, add to -8: -8 + (-4) = -12.
• Multiply -12 * 1 = -12, add to 12: 12 + (-12) = 0.

The quotient is x^3 + 3x^2 - 4x - 12.

Next, factor the cubic quotient further, possibly using rational root theorem. Testing x = 2:

p(2) = 8 + 12 - 8 -12 = 0, so x=2 is a root.

Perform synthetic division again:

• Coefficients: 1 | 3 | -4 | -12
• Bring down 1.
• Multiply 1 * 2 = 2, add to 3: 3 + 2 = 5.
• Multiply 5 * 2 = 10, add to -4: -4 + 10 = 6.
• Multiply 6 * 2 = 12, add to -12: -12 + 12 = 0.

Remaining quotient: x^2 + 5x + 6.

Factor quadratic: x^2 + 5x + 6 = (x + 2)(x + 3).

Thus, the complete factorization of p(x) is:

p(x) = (x - 1)(x - 2)(x + 2)(x + 3).

The zeros are x = 1, 2, -2, -3, where x=1 was given. These roots confirm the roots align with the factors derived from the polynomial.

Conclusion

This mathematical exploration demonstrates the process of algebraic simplifications, vertex form analysis and graphing of quadratics, as well as factorization techniques for quartic polynomials. Each step involves systematic application of algebraic principles and provides insight into the behaviors and properties of polynomial functions.

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