Factor The Trinomial And Write Your Answer As The Product Of Its Two Factors.$x^2 + 2x - 24 = $\square$

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Introduction

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Factoring a trinomial is a fundamental concept in algebra that involves expressing a quadratic expression as the product of two binomials. In this article, we will focus on factoring the trinomial $x^2 + 2x - 24$ and write our answer as the product of its two factors.

What is a Trinomial?

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A trinomial is a polynomial expression that consists of three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In the given trinomial $x^2 + 2x - 24$, the coefficients are $a = 1$, $b = 2$, and $c = -24$.

The Factoring Process

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To factor a trinomial, we need to find two binomials whose product is equal to the original trinomial. The factoring process involves the following steps:

1. Identify the coefficients: Identify the coefficients $a$, $b$, and $c$ in the trinomial.
2. Find the product of the coefficients: Find the product of the coefficients $a$ and $c$, which is $ac$.
3. Find the sum of the coefficients: Find the sum of the coefficients $b$ and $c$, which is $b + c$.
4. Find the two binomials: Find two binomials whose product is equal to the original trinomial, and whose sum of the coefficients is equal to $b + c$.

Factoring the Trinomial

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Now, let's apply the factoring process to the trinomial $x^2 + 2x - 24$.

Step 1: Identify the coefficients

The coefficients in the trinomial are $a = 1$, $b = 2$, and $c = -24$.

Step 2: Find the product of the coefficients

The product of the coefficients $a$ and $c$ is $ac = 1 \times -24 = -24$.

Step 3: Find the sum of the coefficients

The sum of the coefficients $b$ and $c$ is $b + c = 2 + (-24) = -22$.

Step 4: Find the two binomials

We need to find two binomials whose product is equal to the original trinomial, and whose sum of the coefficients is equal to $b + c = -22$. Let's try to find two binomials whose product is equal to $x^2 + 2x - 24$.

After some trial and error, we find that the two binomials are $(x + 6)(x - 4)$.

Verifying the Factored Form

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To verify that the factored form is correct, we need to multiply the two binomials together and check if we get the original trinomial.

$(x + 6)(x - 4) = x^2 - 4x + 6x - 24 = x^2 + 2x - 24$

Yes, we get the original trinomial! Therefore, the factored form of the trinomial $x^2 + 2x - 24$ is $(x + 6)(x - 4)$.

Conclusion

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In this article, we have learned how to factor a trinomial using the factoring process. We have applied the factoring process to the trinomial $x^2 + 2x - 24$ and found that the factored form is $(x + 6)(x - 4)$. We have also verified that the factored form is correct by multiplying the two binomials together.

Frequently Asked Questions

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Q: What is a trinomial?

A: A trinomial is a polynomial expression that consists of three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I factor a trinomial?

A: To factor a trinomial, you need to find two binomials whose product is equal to the original trinomial. The factoring process involves identifying the coefficients, finding the product of the coefficients, finding the sum of the coefficients, and finding the two binomials.

Q: What is the factored form of the trinomial $x^2 + 2x - 24$?

A: The factored form of the trinomial $x^2 + 2x - 24$ is $(x + 6)(x - 4)$.

References

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• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Further Reading

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• [1] "Factoring Quadratic Expressions" by Math Open Reference
• [2] "Factoring Trinomials" by Purplemath
• [3] "Factoring Polynomials" by Khan Academy
  

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Introduction

---

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as the product of two binomials. In this article, we will provide a comprehensive Q&A guide on factoring trinomials, covering common questions and answers, examples, and tips.

Q&A

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Q: What is a trinomial?

A: A trinomial is a polynomial expression that consists of three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I factor a trinomial?

A: To factor a trinomial, you need to find two binomials whose product is equal to the original trinomial. The factoring process involves identifying the coefficients, finding the product of the coefficients, finding the sum of the coefficients, and finding the two binomials.

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a quadratic expression as the product of two binomials, while simplifying involves combining like terms to reduce the expression to its simplest form.

Q: Can I factor a trinomial with a negative coefficient?

A: Yes, you can factor a trinomial with a negative coefficient. The process is the same as factoring a trinomial with a positive coefficient.

Q: How do I factor a trinomial with a coefficient of 1?

A: When the coefficient of the quadratic term is 1, you can factor the trinomial by finding two binomials whose product is equal to the original trinomial.

Q: Can I factor a trinomial with a coefficient of 0?

A: No, you cannot factor a trinomial with a coefficient of 0. This is because the product of two binomials cannot be equal to 0.

Q: How do I factor a trinomial with a coefficient of -1?

A: When the coefficient of the quadratic term is -1, you can factor the trinomial by finding two binomials whose product is equal to the original trinomial.

Q: Can I factor a trinomial with a coefficient of 1 and a negative constant term?

A: Yes, you can factor a trinomial with a coefficient of 1 and a negative constant term. The process is the same as factoring a trinomial with a positive coefficient.

Q: How do I factor a trinomial with a coefficient of -1 and a negative constant term?

A: When the coefficient of the quadratic term is -1 and the constant term is negative, you can factor the trinomial by finding two binomials whose product is equal to the original trinomial.

Examples

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Example 1: Factoring a Trinomial with a Positive Coefficient

Factor the trinomial $x^2 + 5x + 6$.

Solution: $(x + 2)(x + 3)$

Example 2: Factoring a Trinomial with a Negative Coefficient

Factor the trinomial $x^2 - 3x - 4$.

Solution: $(x - 4)(x + 1)$

Example 3: Factoring a Trinomial with a Coefficient of 1

Factor the trinomial $x^2 + 2x + 1$.

Solution: $(x + 1)^2$

Example 4: Factoring a Trinomial with a Coefficient of -1

Factor the trinomial $x^2 - 2x - 3$.

Solution: $(x - 3)(x + 1)$

Tips

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Tip 1: Use the Factoring Process

To factor a trinomial, use the factoring process: identify the coefficients, find the product of the coefficients, find the sum of the coefficients, and find the two binomials.

Tip 2: Look for Common Factors

Look for common factors in the trinomial, such as a common factor of 2 or 3.

Tip 3: Use the Distributive Property

Use the distributive property to expand the two binomials and check if they are equal to the original trinomial.

Tip 4: Check Your Work

Check your work by multiplying the two binomials together and checking if they are equal to the original trinomial.

Conclusion

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Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as the product of two binomials. In this article, we have provided a comprehensive Q&A guide on factoring trinomials, covering common questions and answers, examples, and tips. We hope that this guide has been helpful in understanding the concept of factoring trinomials.

Frequently Asked Questions

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Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a quadratic expression as the product of two binomials, while simplifying involves combining like terms to reduce the expression to its simplest form.

Q: Can I factor a trinomial with a negative coefficient?

A: Yes, you can factor a trinomial with a negative coefficient. The process is the same as factoring a trinomial with a positive coefficient.

Q: How do I factor a trinomial with a coefficient of 1?

A: When the coefficient of the quadratic term is 1, you can factor the trinomial by finding two binomials whose product is equal to the original trinomial.

Q: Can I factor a trinomial with a coefficient of 0?

A: No, you cannot factor a trinomial with a coefficient of 0. This is because the product of two binomials cannot be equal to 0.

References

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• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Further Reading

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• [1] "Factoring Quadratic Expressions" by Math Open Reference
• [2] "Factoring Trinomials" by Purplemath
• [3] "Factoring Polynomials" by Khan Academy