What Is The Factorization Of The Polynomial Below?$2x^2 + 28x + 98$A. $(2x + 7)(x + 7$\]B. $(x + 7)(x + 2$\]C. $(x + 7)(x + 14$\]D. $2(x + 7)(x + 7$\]

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Understanding Polynomial Factorization

Polynomial factorization is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This process is essential in solving equations, finding roots, and simplifying expressions. In this article, we will focus on factorizing the given polynomial $2x^2 + 28x + 98$.

The Given Polynomial

The given polynomial is $2x^2 + 28x + 98$. To factorize this polynomial, we need to find two binomials whose product equals the given polynomial. The general form of a quadratic polynomial is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Factoring the Polynomial

To factorize the polynomial $2x^2 + 28x + 98$, we need to find two binomials whose product equals the given polynomial. We can start by looking for common factors. In this case, the greatest common factor (GCF) of the coefficients $2$, $28$, and $98$ is $2$. We can factor out the GCF as follows:

$$2x^2 + 28x + 98 = 2(x^2 + 14x + 49)$$

Factoring the Quadratic Expression

Now, we need to factorize the quadratic expression $x^2 + 14x + 49$. We can start by looking for two numbers whose product equals the constant term $49$ and whose sum equals the coefficient of the linear term $14$. These numbers are $7$ and $7$, since $7 \times 7 = 49$ and $7 + 7 = 14$.

Factoring the Quadratic Expression (continued)

Using the numbers $7$ and $7$, we can rewrite the quadratic expression as follows:

$$x^2 + 14x + 49 = (x + 7)(x + 7)$$

Factoring the Polynomial (continued)

Now, we can substitute the factored quadratic expression back into the original polynomial:

$$2x^2 + 28x + 98 = 2(x + 7)(x + 7)$$

Conclusion

In conclusion, the factorization of the polynomial $2x^2 + 28x + 98$ is $2(x + 7)(x + 7)$. This factorization can be verified by multiplying the two binomials together to obtain the original polynomial.

Comparison with Other Options

Let's compare our factorization with the other options provided:

• Option A: $(2x + 7)(x + 7)$
• Option B: $(x + 7)(x + 2)$
• Option C: $(x + 7)(x + 14)$
• Option D: $2(x + 7)(x + 7)$

Our factorization matches option D, which is $2(x + 7)(x + 7)$.

Importance of Polynomial Factorization

Polynomial factorization is an essential concept in algebra that has numerous applications in various fields, including mathematics, science, and engineering. It is used to solve equations, find roots, and simplify expressions. In this article, we have demonstrated how to factorize a polynomial using the method of factoring out the greatest common factor and then factoring the resulting quadratic expression.

Real-World Applications of Polynomial Factorization

Polynomial factorization has numerous real-world applications, including:

• Solving Equations: Polynomial factorization is used to solve equations by finding the roots of the polynomial.
• Finding Roots: Polynomial factorization is used to find the roots of a polynomial, which is essential in various fields, including physics and engineering.
• Simplifying Expressions: Polynomial factorization is used to simplify expressions by combining like terms.

Conclusion

In conclusion, polynomial factorization is a fundamental concept in algebra that has numerous applications in various fields. In this article, we have demonstrated how to factorize a polynomial using the method of factoring out the greatest common factor and then factoring the resulting quadratic expression. We have also compared our factorization with other options and highlighted the importance of polynomial factorization in various fields.

Final Answer

The final answer is: $\boxed{2(x + 7)(x + 7)}$

Understanding Polynomial Factorization

Polynomial factorization is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This process is essential in solving equations, finding roots, and simplifying expressions. In this article, we will provide answers to frequently asked questions about polynomial factorization.

Q: What is polynomial factorization?

A: Polynomial factorization is the process of expressing a polynomial as a product of simpler polynomials.

Q: Why is polynomial factorization important?

A: Polynomial factorization is essential in solving equations, finding roots, and simplifying expressions. It has numerous applications in various fields, including mathematics, science, and engineering.

Q: How do I factorize a polynomial?

A: To factorize a polynomial, you need to find two binomials whose product equals the given polynomial. You can start by looking for common factors and then factorizing the resulting quadratic expression.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides all the terms of a polynomial.

Q: How do I find the GCF of a polynomial?

A: To find the GCF of a polynomial, you need to list all the factors of each term and then find the largest factor that is common to all the terms.

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves combining like terms to reduce the complexity of an expression.

Q: Can you provide an example of polynomial factorization?

A: Yes, let's consider the polynomial $2x^2 + 28x + 98$. To factorize this polynomial, we need to find two binomials whose product equals the given polynomial. We can start by looking for common factors and then factorizing the resulting quadratic expression.

Q: How do I know if a polynomial can be factored?

A: A polynomial can be factored if it can be expressed as a product of simpler polynomials. You can use various methods, including factoring out the greatest common factor and then factorizing the resulting quadratic expression.

Q: What are some common mistakes to avoid when factoring polynomials?

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

• Not factoring out the greatest common factor
• Not using the correct method for factoring quadratic expressions
• Not checking the result for accuracy

Q: Can you provide some tips for factoring polynomials?

A: Yes, here are some tips for factoring polynomials:

• Start by looking for common factors
• Use the correct method for factoring quadratic expressions
• Check the result for accuracy
• Use a calculator or computer software to check your work

Q: How do I know if I have factored a polynomial correctly?

A: To check if you have factored a polynomial correctly, you need to multiply the two binomials together to obtain the original polynomial. If the result is the same as the original polynomial, then you have factored it correctly.

Q: What are some real-world applications of polynomial factorization?

A: Polynomial factorization has numerous real-world applications, including:

• Solving equations
• Finding roots
• Simplifying expressions
• Modeling real-world phenomena

Q: Can you provide some examples of polynomial factorization in real-world applications?

A: Yes, here are some examples of polynomial factorization in real-world applications:

• Solving equations in physics and engineering
• Finding roots in computer science and cryptography
• Simplifying expressions in finance and economics
• Modeling real-world phenomena in biology and medicine

Conclusion

In conclusion, polynomial factorization is a fundamental concept in algebra that has numerous applications in various fields. In this article, we have provided answers to frequently asked questions about polynomial factorization, including how to factorize a polynomial, what is the greatest common factor, and how to check if you have factored a polynomial correctly. We have also provided some tips for factoring polynomials and some examples of polynomial factorization in real-world applications.