Which Expression Is Equivalent To The Polynomial 16 X 2 + 4 16x^2 + 4 16 X 2 + 4 ?A. ( 4 X + 2 I ) ( 4 X − 2 I (4x + 2i)(4x - 2i ( 4 X + 2 I ) ( 4 X − 2 I ]B. ( 4 X + 2 ) ( 4 X − 2 (4x + 2)(4x - 2 ( 4 X + 2 ) ( 4 X − 2 ]C. ( 4 X + 2 ) 2 (4x + 2)^2 ( 4 X + 2 ) 2 D. ( 4 X − 2 I ) 2 (4x - 2i)^2 ( 4 X − 2 I ) 2

Which Expression is Equivalent to the Polynomial $16x^2 + 4$?

In algebra, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When it comes to simplifying or factoring polynomials, we often look for equivalent expressions that can help us solve equations or perform other mathematical operations. In this article, we will explore which expression is equivalent to the polynomial $16x^2 + 4$.

The given polynomial is $16x^2 + 4$. To find an equivalent expression, we need to analyze its structure and identify any patterns or relationships that can help us simplify it. The polynomial consists of a quadratic term $16x^2$ and a constant term $4$.

One way to simplify the polynomial is to factor it. Factoring involves expressing a polynomial as a product of simpler polynomials. In this case, we can try to factor the quadratic term $16x^2$.

16x^2 = (4x)^2

Now, we can see that the quadratic term is a perfect square trinomial, which can be factored as:

(4x)^2 = (4x)(4x)

However, this is not one of the answer choices. We need to find a way to incorporate the constant term $4$ into the factored expression.

The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. We can use this formula to simplify the polynomial.

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

Now, we can apply the difference of squares formula:

(4x)^2 + 2^2 = (4x + 2)(4x - 2)

This expression is equivalent to the original polynomial $16x^2 + 4$.

Now that we have found an equivalent expression for the polynomial, let's evaluate the answer choices:

A. $(4x + 2i)(4x - 2i)$: This expression is not equivalent to the polynomial $16x^2 + 4$ because it contains complex numbers.

B. $(4x + 2)(4x - 2)$: This expression is equivalent to the polynomial $16x^2 + 4$.

C. $(4x + 2)^2$: This expression is not equivalent to the polynomial $16x^2 + 4$ because it does not include the constant term $4$.

D. $(4x - 2i)^2$: This expression is not equivalent to the polynomial $16x^2 + 4$ because it contains complex numbers.

In conclusion, the expression $(4x + 2)(4x - 2)$ is equivalent to the polynomial $16x^2 + 4$. This expression can be obtained by factoring the quadratic term $16x^2$ and using the difference of squares formula to simplify the polynomial.

The final answer is B. $(4x + 2)(4x - 2)$.
Q&A: Which Expression is Equivalent to the Polynomial $16x^2 + 4$?

In our previous article, we explored which expression is equivalent to the polynomial $16x^2 + 4$. We found that the expression $(4x + 2)(4x - 2)$ is equivalent to the polynomial. In this article, we will answer some frequently asked questions related to this topic.

A: The expressions $(4x + 2)(4x - 2)$ and $(4x + 2)^2$ are not equivalent. The expression $(4x + 2)(4x - 2)$ is a difference of squares, while the expression $(4x + 2)^2$ is a perfect square trinomial.

A: The expression $(4x + 2i)(4x - 2i)$ is not equivalent to the polynomial $16x^2 + 4$ because it contains complex numbers. The polynomial $16x^2 + 4$ is a real polynomial, and it does not have any complex roots.

A: Yes, we can factor the polynomial $16x^2 + 4$ using other methods. One way to factor it is to use the method of completing the square. This method involves rewriting the polynomial in a form that allows us to easily factor it.

A: The difference of squares formula is a key concept in this problem. It allows us to simplify the polynomial $16x^2 + 4$ by expressing it as a product of two binomials. This formula is a powerful tool for factoring polynomials and solving equations.

A: Yes, we can use the expression $(4x + 2)(4x - 2)$ to solve equations involving the polynomial $16x^2 + 4$. This expression can be used to find the roots of the polynomial, which can then be used to solve equations involving the polynomial.

A: Some common mistakes to avoid when factoring polynomials include:

• Not checking if the polynomial can be factored using the difference of squares formula
• Not using the correct method to factor the polynomial
• Not checking if the factored expression is equivalent to the original polynomial

In conclusion, the expression $(4x + 2)(4x - 2)$ is equivalent to the polynomial $16x^2 + 4$. We hope that this Q&A article has provided some helpful insights and answers to frequently asked questions related to this topic.

The final answer is B. $(4x + 2)(4x - 2)$.