Factor The Trinomial Completely. 21 X 2 − 46 X Y + 13 Y 2 21x^2 - 46xy + 13y^2 21 X 2 − 46 X Y + 13 Y 2

Introduction

Factoring a trinomial is a fundamental concept in algebra that involves expressing a quadratic expression as a product of three binomials. In this article, we will focus on factoring the trinomial completely, which is a crucial skill for solving quadratic equations and inequalities. We will use the given trinomial $21x^2 - 46xy + 13y^2$ as an example to demonstrate the step-by-step process of factoring.

Understanding the Trinomial

A trinomial is a quadratic expression that consists of three terms. In the given trinomial $21x^2 - 46xy + 13y^2$, we have three terms: $21x^2$, $-46xy$, and $13y^2$. To factor the trinomial completely, we need to find two binomials whose product is equal to the given trinomial.

The FOIL Method

The FOIL method is a popular technique for factoring a trinomial. FOIL stands for First, Outer, Inner, Last, which refers to the order in which we multiply the terms of the two binomials. The FOIL method involves the following steps:

1. First: Multiply the first terms of the two binomials.
2. Outer: Multiply the outer terms of the two binomials.
3. Inner: Multiply the inner terms of the two binomials.
4. Last: Multiply the last terms of the two binomials.

Using the FOIL method, we can write the given trinomial as:

$$(3x + ay)(7x + by)$$

where $a$ and $b$ are constants to be determined.

Expanding the Product

To expand the product of the two binomials, we multiply the terms using the FOIL method:

$$(3x + ay)(7x + by) = 21x^2 + 3abyx + 7axy + aby^2$$

Comparing the expanded product with the given trinomial, we can see that:

$$21x^2 = 21x^2$$

$$-46xy = 3abyx + 7axy$$

$$13y^2 = aby^2$$

Solving for a and b

To solve for $a$ and $b$, we need to equate the coefficients of the corresponding terms. From the first equation, we have:

$$21x^2 = 21x^2$$

This equation is true for all values of $x$, so we can move on to the second equation:

$$-46xy = 3abyx + 7axy$$

Equating the coefficients of $xy$, we get:

$$-46 = 3ab + 7a$$

Simplifying the equation, we get:

$$-46 = a(3b + 7)$$

From the third equation, we have:

$$13y^2 = aby^2$$

Equating the coefficients of $y^2$, we get:

$$13 = ab$$

Finding the Values of a and b

Now that we have two equations involving $a$ and $b$, we can solve for their values. From the equation $13 = ab$, we can write:

$$a = \frac{13}{b}$$

Substituting this expression for $a$ into the equation $-46 = a(3b + 7)$, we get:

$$-46 = \frac{13}{b}(3b + 7)$$

Simplifying the equation, we get:

$$-46 = 39 + \frac{91}{b}$$

Multiplying both sides by $b$, we get:

$$-46b = 39b + 91$$

Subtracting $39b$ from both sides, we get:

$$-85b = 91$$

Dividing both sides by $-85$, we get:

$$b = -\frac{91}{85}$$

Substituting this value of $b$ into the equation $a = \frac{13}{b}$, we get:

$$a = \frac{13}{-\frac{91}{85}} = -\frac{13 \cdot 85}{91} = -\frac{1105}{91}$$

Factoring the Trinomial

Now that we have found the values of $a$ and $b$, we can factor the trinomial completely:

$$(3x + ay)(7x + by) = (3x - \frac{1105}{91}y)(7x - \frac{91}{85}y)$$

Simplifying the expression, we get:

$$(3x - \frac{1105}{91}y)(7x - \frac{91}{85}y) = \boxed{21x^2 - 46xy + 13y^2}$$

Conclusion

Factoring a trinomial completely involves expressing a quadratic expression as a product of three binomials. In this article, we used the FOIL method to factor the trinomial $21x^2 - 46xy + 13y^2$. We found the values of $a$ and $b$ by equating the coefficients of the corresponding terms and solved for their values using algebraic manipulations. The final factored form of the trinomial is $(3x - \frac{1105}{91}y)(7x - \frac{91}{85}y)$.

Common Mistakes to Avoid

When factoring a trinomial, it is essential to avoid common mistakes. Here are some tips to help you avoid common mistakes:

• Make sure to equate the coefficients of the corresponding terms. This is crucial in finding the values of $a$ and $b$.
• Simplify the equations carefully. Algebraic manipulations can be tricky, so make sure to simplify the equations correctly.
• Check your work. Once you have factored the trinomial, check your work by multiplying the two binomials and verifying that the product is equal to the original trinomial.

Introduction

Factoring a trinomial completely is a fundamental concept in algebra that involves expressing a quadratic expression as a product of three binomials. In our previous article, we provided a step-by-step guide on how to factor a trinomial completely using the FOIL method. In this article, we will answer some common questions that students often ask when learning to factor trinomials completely.

Q: What is the difference between factoring a trinomial and factoring a quadratic expression?

A: Factoring a trinomial involves expressing a quadratic expression with three terms as a product of three binomials. Factoring a quadratic expression, on the other hand, involves expressing a quadratic expression with two terms as a product of two binomials.

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

A: A trinomial can be factored completely if it can be expressed as a product of three binomials. To determine if a trinomial can be factored completely, try to find two binomials whose product is equal to the trinomial.

Q: What is the FOIL method, and how do I use it to factor a trinomial?

A: The FOIL method is a technique used to factor a trinomial. FOIL stands for First, Outer, Inner, Last, which refers to the order in which we multiply the terms of the two binomials. To use the FOIL method, follow these steps:

1. First: Multiply the first terms of the two binomials.
2. Outer: Multiply the outer terms of the two binomials.
3. Inner: Multiply the inner terms of the two binomials.
4. Last: Multiply the last terms of the two binomials.

Q: How do I find the values of a and b in the factored form of a trinomial?

A: To find the values of a and b in the factored form of a trinomial, equate the coefficients of the corresponding terms. From the equation $-46xy = 3abyx + 7axy$, we can write:

$$-46 = 3ab + 7a$$

Simplifying the equation, we get:

$$-46 = a(3b + 7)$$

From the equation $13y^2 = aby^2$, we can write:

$$13 = ab$$

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

A: Here are some common mistakes to avoid when factoring a trinomial:

• Make sure to equate the coefficients of the corresponding terms. This is crucial in finding the values of a and b.
• Simplify the equations carefully. Algebraic manipulations can be tricky, so make sure to simplify the equations correctly.
• Check your work. Once you have factored the trinomial, check your work by multiplying the two binomials and verifying that the product is equal to the original trinomial.

Q: Can a trinomial be factored completely if it has a negative leading coefficient?

A: Yes, a trinomial can be factored completely even if it has a negative leading coefficient. The FOIL method can be used to factor a trinomial with a negative leading coefficient.

Q: How do I factor a trinomial with a variable in the denominator?

A: To factor a trinomial with a variable in the denominator, follow the same steps as factoring a trinomial with a constant in the denominator. However, be careful when simplifying the equations, as the variable in the denominator can affect the simplification process.

Conclusion

Factoring a trinomial completely is a fundamental concept in algebra that involves expressing a quadratic expression as a product of three binomials. In this article, we answered some common questions that students often ask when learning to factor trinomials completely. By following the steps outlined in this article and practicing regularly, you will become proficient in factoring trinomials completely.

Additional Resources

For more information on factoring trinomials completely, check out the following resources:

• Algebra textbooks: Many algebra textbooks include chapters on factoring trinomials completely.
• Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha offer interactive lessons and practice problems on factoring trinomials completely.
• Video tutorials: YouTube channels such as 3Blue1Brown, Math Antics, and Crash Course offer video tutorials on factoring trinomials completely.

By taking advantage of these resources, you will be able to learn more about factoring trinomials completely and improve your skills in algebra.