Select The Correct Answer.Which Expression Is Equivalent To This Polynomial? $16x^2 + 4$A. $(4x + 2i)(4x - 2i)$B. $ ( 4 X + 2 ) ( 4 X − 2 ) (4x + 2)(4x - 2) ( 4 X + 2 ) ( 4 X − 2 ) [/tex]C. $(4x + 2)^2$D. $(4x - 2i)^2$

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Introduction

Polynomial equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of polynomial equation, and we will explore the different methods and techniques used to find the correct solution.

Understanding the Problem

The problem we are given is a quadratic polynomial equation in the form of 16x2+416x^2 + 4. Our task is to find an equivalent expression for this polynomial. We are presented with four options, and we need to determine which one is correct.

Option A: (4x+2i)(4x−2i)(4x + 2i)(4x - 2i)

Let's start by analyzing the first option, which is (4x+2i)(4x−2i)(4x + 2i)(4x - 2i). To determine if this is the correct solution, we need to multiply the two binomials using the FOIL method.

FOIL Method

The FOIL method is a technique used to multiply two binomials. It stands for First, Outer, Inner, Last, and it refers to the order in which we multiply the terms.

  • First: Multiply the first terms of each binomial.
  • Outer: Multiply the outer terms of each binomial.
  • Inner: Multiply the inner terms of each binomial.
  • Last: Multiply the last terms of each binomial.

Using the FOIL method, we get:

(4x+2i)(4x−2i)=16x2−8xi+8xi−4i2(4x + 2i)(4x - 2i) = 16x^2 - 8xi + 8xi - 4i^2

Since i2=−1i^2 = -1, we can simplify the expression:

16x2−8xi+8xi+4=16x2+416x^2 - 8xi + 8xi + 4 = 16x^2 + 4

This is the same expression we were given in the problem. Therefore, option A is a possible solution.

Option B: (4x+2)(4x−2)(4x + 2)(4x - 2)

Let's analyze the second option, which is (4x+2)(4x−2)(4x + 2)(4x - 2). To determine if this is the correct solution, we need to multiply the two binomials using the FOIL method.

Using the FOIL method, we get:

(4x+2)(4x−2)=16x2−8x+8x−4(4x + 2)(4x - 2) = 16x^2 - 8x + 8x - 4

Simplifying the expression, we get:

16x2−416x^2 - 4

This is not the same expression we were given in the problem. Therefore, option B is not a possible solution.

Option C: (4x+2)2(4x + 2)^2

Let's analyze the third option, which is (4x+2)2(4x + 2)^2. To determine if this is the correct solution, we need to expand the squared binomial.

Using the formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2, we get:

(4x+2)2=16x2+16x+4(4x + 2)^2 = 16x^2 + 16x + 4

This is not the same expression we were given in the problem. Therefore, option C is not a possible solution.

Option D: (4x−2i)2(4x - 2i)^2

Let's analyze the fourth option, which is (4x−2i)2(4x - 2i)^2. To determine if this is the correct solution, we need to expand the squared binomial.

Using the formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2, we get:

(4x−2i)2=16x2−16xi+4i2(4x - 2i)^2 = 16x^2 - 16xi + 4i^2

Since i2=−1i^2 = -1, we can simplify the expression:

16x2−16xi−416x^2 - 16xi - 4

This is not the same expression we were given in the problem. Therefore, option D is not a possible solution.

Conclusion

After analyzing all the options, we can conclude that the correct solution is option A, which is (4x+2i)(4x−2i)(4x + 2i)(4x - 2i). This expression is equivalent to the given polynomial 16x2+416x^2 + 4.

Final Answer

The final answer is:

  • Option A: (4x+2i)(4x−2i)(4x + 2i)(4x - 2i)

This is the correct solution to the problem.

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Q: What is a polynomial equation?

A polynomial equation is an algebraic equation in which the highest power of the variable (usually x) is a non-negative integer. In other words, it is an equation in which the variable is raised to a power that is a whole number.

Q: What are the different types of polynomial equations?

There are several types of polynomial equations, including:

  • Linear equations: These are polynomial equations in which the highest power of the variable is 1. For example, 2x + 3 = 0 is a linear equation.
  • Quadratic equations: These are polynomial equations in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.
  • Cubic equations: These are polynomial equations in which the highest power of the variable is 3. For example, x^3 + 2x^2 + x + 1 = 0 is a cubic equation.
  • Quartic equations: These are polynomial equations in which the highest power of the variable is 4. For example, x^4 + 3x^3 + 2x^2 + x + 1 = 0 is a quartic equation.

Q: How do I solve a polynomial equation?

There are several methods for solving polynomial equations, including:

  • Factoring: This involves expressing the polynomial as a product of simpler polynomials. For example, x^2 + 4x + 4 = (x + 2)^2.
  • Quadratic formula: This involves using the quadratic formula to find the solutions to a quadratic equation. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
  • Cubic formula: This involves using the cubic formula to find the solutions to a cubic equation. The cubic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
  • Quartic formula: This involves using the quartic formula to find the solutions to a quartic equation. The quartic formula is x = (-b ± √(b^2 - 4ac)) / 2a.

Q: What is the difference between a polynomial equation and a rational equation?

A polynomial equation is an equation in which the variable is raised to a power that is a whole number. A rational equation, on the other hand, is an equation in which the variable is raised to a power that is a fraction or a decimal.

Q: How do I simplify a polynomial equation?

There are several methods for simplifying a polynomial equation, including:

  • Combining like terms: This involves combining terms that have the same variable and exponent. For example, 2x^2 + 3x^2 = 5x^2.
  • Factoring: This involves expressing the polynomial as a product of simpler polynomials. For example, x^2 + 4x + 4 = (x + 2)^2.
  • Using the distributive property: This involves multiplying each term in the polynomial by a constant or a variable. For example, 2(x^2 + 3x + 4) = 2x^2 + 6x + 8.

Q: What is the difference between a polynomial equation and a trigonometric equation?

A polynomial equation is an equation in which the variable is raised to a power that is a whole number. A trigonometric equation, on the other hand, is an equation that involves trigonometric functions such as sine, cosine, and tangent.

Q: How do I solve a trigonometric equation?

There are several methods for solving trigonometric equations, including:

  • Using the unit circle: This involves using the unit circle to find the values of trigonometric functions.
  • Using trigonometric identities: This involves using trigonometric identities such as the Pythagorean identity to simplify the equation.
  • Using algebraic methods: This involves using algebraic methods such as factoring and combining like terms to solve the equation.

Q: What is the difference between a polynomial equation and a differential equation?

A polynomial equation is an equation in which the variable is raised to a power that is a whole number. A differential equation, on the other hand, is an equation that involves the derivative of a function.

Q: How do I solve a differential equation?

There are several methods for solving differential equations, including:

  • Using separation of variables: This involves separating the variables in the equation and integrating each side separately.
  • Using integration factors: This involves using integration factors to simplify the equation.
  • Using numerical methods: This involves using numerical methods such as the Euler method to approximate the solution to the equation.

Conclusion

In this article, we have discussed some of the most frequently asked questions about solving polynomial equations. We have covered topics such as the different types of polynomial equations, methods for solving polynomial equations, and the difference between polynomial equations and other types of equations. We hope that this article has been helpful in answering some of your questions about solving polynomial equations.