What Is The Factored Form Of The Polynomial $x^2 - 16x + 48$?A. $(x-4)(x-12$\] B. $(x-6)(x-8$\] C. $(x+4)(x+12$\] D. $(x+6)(x+8$\]

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Introduction

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Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring the polynomial $x^2 - 16x + 48$ and explore the different methods used to achieve this.

What is Factoring?

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Factoring a polynomial involves expressing it as a product of two or more polynomials. This can be done in various ways, including:

• Factoring out the greatest common factor (GCF): This involves identifying the largest factor that divides all the terms of the polynomial.
• Factoring by grouping: This involves grouping the terms of the polynomial into pairs and factoring out the common factors from each pair.
• Factoring quadratic expressions: This involves expressing a quadratic expression in the form of $(x-a)(x-b)$, where $a$ and $b$ are the roots of the quadratic equation.

Factoring the Polynomial $x^2 - 16x + 48$

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To factor the polynomial $x^2 - 16x + 48$, we can use the method of factoring quadratic expressions. This involves expressing the polynomial in the form of $(x-a)(x-b)$, where $a$ and $b$ are the roots of the quadratic equation.

Step 1: Identify the Coefficients

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The given polynomial is $x^2 - 16x + 48$. The coefficients of this polynomial are:

• Coefficient of $x^2$: 1
• Coefficient of $x$: -16
• Constant term: 48

Step 2: Find the Roots

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To find the roots of the quadratic equation, we can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -16$, and $c = 48$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(48)}}{2(1)}$$

Simplifying this expression, we get:

$$x = \frac{16 \pm \sqrt{256 - 192}}{2}$$

$$x = \frac{16 \pm \sqrt{64}}{2}$$

$$x = \frac{16 \pm 8}{2}$$

Therefore, the roots of the quadratic equation are:

$$x = \frac{16 + 8}{2} = 12$$

$$x = \frac{16 - 8}{2} = 4$$

Step 3: Write the Factored Form

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Now that we have found the roots of the quadratic equation, we can write the factored form of the polynomial:

$$(x-4)(x-12)$$

This is the factored form of the polynomial $x^2 - 16x + 48$.

Conclusion

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In this article, we have explored the concept of factoring polynomials and applied it to the polynomial $x^2 - 16x + 48$. We have used the method of factoring quadratic expressions to express the polynomial in the form of $(x-a)(x-b)$, where $a$ and $b$ are the roots of the quadratic equation. The factored form of the polynomial is $(x-4)(x-12)$.

Answer

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The correct answer is:

• A. $(x-4)(x-12)$

This is the factored form of the polynomial $x^2 - 16x + 48$.

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Introduction

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Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will provide a Q&A guide to help you understand the concept of factoring polynomials and how to apply it to different types of polynomials.

Q&A

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Q: What is factoring?

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A: Factoring a polynomial involves expressing it as a product of two or more polynomials. This can be done in various ways, including factoring out the greatest common factor (GCF), factoring by grouping, and factoring quadratic expressions.

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

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A: The greatest common factor (GCF) is the largest factor that divides all the terms of a polynomial. For example, in the polynomial $x^2 + 5x + 6$, the GCF is $x + 1$.

Q: How do I factor a polynomial by grouping?

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A: To factor a polynomial by grouping, you need to group the terms of the polynomial into pairs and factor out the common factors from each pair. For example, in the polynomial $x^2 + 4x + 4x + 16$, you can group the terms as follows:

• $(x^2 + 4x)$
• $(4x + 16)$

Then, you can factor out the common factors from each pair:

• $x(x + 4)$
• $4(x + 4)$

Finally, you can combine the two expressions to get the factored form of the polynomial:

$(x + 4)(x + 4)$

Q: How do I factor a quadratic expression?

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A: To factor a quadratic expression, you need to express it in the form of $(x-a)(x-b)$, where $a$ and $b$ are the roots of the quadratic equation. For example, in the quadratic expression $x^2 + 5x + 6$, you can factor it as follows:

$(x + 3)(x + 2)$

Q: What is the difference between factoring and simplifying?

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A: Factoring and simplifying are two different concepts in algebra. Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves combining like terms to get a simpler expression.

Q: Can you provide an example of factoring a polynomial?

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A: Yes, let's consider the polynomial $x^2 - 16x + 48$. To factor this polynomial, we can use the method of factoring quadratic expressions. We can express the polynomial in the form of $(x-a)(x-b)$, where $a$ and $b$ are the roots of the quadratic equation.

The roots of the quadratic equation are $x = 4$ and $x = 12$. Therefore, the factored form of the polynomial is:

$(x - 4)(x - 12)$

Q: Can you provide an example of factoring a polynomial by grouping?

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A: Yes, let's consider the polynomial $x^2 + 4x + 4x + 16$. To factor this polynomial, we can use the method of factoring by grouping. We can group the terms of the polynomial into pairs and factor out the common factors from each pair.

The factored form of the polynomial is:

$(x + 4)(x + 4)$

Q: Can you provide an example of factoring a quadratic expression?

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A: Yes, let's consider the quadratic expression $x^2 + 5x + 6$. To factor this expression, we can use the method of factoring quadratic expressions. We can express the expression in the form of $(x-a)(x-b)$, where $a$ and $b$ are the roots of the quadratic equation.

The roots of the quadratic equation are $x = -3$ and $x = -2$. Therefore, the factored form of the expression is:

$(x + 3)(x + 2)$

Conclusion

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In this article, we have provided a Q&A guide to help you understand the concept of factoring polynomials and how to apply it to different types of polynomials. We have covered topics such as factoring out the greatest common factor (GCF), factoring by grouping, and factoring quadratic expressions. We have also provided examples of factoring polynomials using different methods.

Answer

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The correct answers to the Q&A guide are:

• Q: What is factoring? A: Factoring a polynomial involves expressing it as a product of two or more polynomials.
• 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 factor a polynomial by grouping? A: To factor a polynomial by grouping, you need to group the terms of the polynomial into pairs and factor out the common factors from each pair.
• Q: How do I factor a quadratic expression? A: To factor a quadratic expression, you need to express it in the form of $(x-a)(x-b)$, where $a$ and $b$ are the roots of the quadratic equation.
• Q: What is the difference between factoring and simplifying? A: Factoring and simplifying are two different concepts in algebra. Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves combining like terms to get a simpler expression.
• Q: Can you provide an example of factoring a polynomial? A: Yes, let's consider the polynomial $x^2 - 16x + 48$. The factored form of the polynomial is $(x - 4)(x - 12)$.
• Q: Can you provide an example of factoring a polynomial by grouping? A: Yes, let's consider the polynomial $x^2 + 4x + 4x + 16$. The factored form of the polynomial is $(x + 4)(x + 4)$.
• Q: Can you provide an example of factoring a quadratic expression? A: Yes, let's consider the quadratic expression $x^2 + 5x + 6$. The factored form of the expression is $(x + 3)(x + 2)$.