Which Of The Following Is A Correct Factorization Of This Trinomial?$-3x^2 - 10x - 8$A. $-(x+4)(x-3$\]B. $-3(x+4)(x+2$\]C. $(-3x+4)(x-2$\]D. $-(3x+4)(x+2$\]

Feb 26, 2025 by ADMIN 157 views

**Factoring Trinomials: A Step-by-Step Guide**

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Understanding Trinomials

A trinomial is a polynomial expression consisting of three terms. It can be written in the form of $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. Factoring trinomials involves expressing them as a product of two binomials.

The Process of Factoring Trinomials

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

Identify the coefficients: Identify the coefficients of the trinomial, which are the numbers in front of the variable terms.
Determine the signs: Determine the signs of the coefficients. If the coefficient of the $x^2$ term is positive, the signs of the two binomials must be the same. If the coefficient of the $x^2$ term is negative, the signs of the two binomials must be different.
Find the factors: Find the factors of the constant term that add up to the coefficient of the $x$ term.
Write the factored form: Write the factored form of the trinomial by multiplying the two binomials.

Factoring the Given Trinomial

The given trinomial is $-3x^2 - 10x - 8$ . To factor this trinomial, we need to follow the steps outlined above.

Step 1: Identify the Coefficients

The coefficients of the trinomial are $-3$ , $-10$ , and $-8$ .

Step 2: Determine the Signs

Since the coefficient of the $x^2$ term is negative, the signs of the two binomials must be different.

Step 3: Find the Factors

We need to find the factors of $-8$ that add up to $-10$ . The factors of $-8$ are $-1$ , $-2$ , $-4$ , and $-8$ . We can see that $-4$ and $-2$ add up to $-6$ , which is not equal to $-10$ . However, $-8$ and $-1$ add up to $-9$ , which is also not equal to $-10$ . But, $-4$ and $-2$ are the closest pair of factors that add up to $-6$ , which is close to $-10$ . We can try to find the correct pair of factors by trial and error.

Step 4: Write the Factored Form

After trying different pairs of factors, we find that $-8$ and $-1$ are not the correct pair, but $-4$ and $-2$ are close. We can try to multiply $-4$ and $-2$ by a constant to get $-10$ . We find that $-4$ and $-2$ multiplied by $-2.5$ equals $-10$ . Therefore, the correct pair of factors is $-4$ and $-2.5$ . However, we cannot have a fraction as a factor, so we need to multiply both factors by $2$ to get rid of the fraction. This gives us $-8$ and $-5$ . Therefore, the factored form of the trinomial is $-(x+4)(x+2)$ .

Conclusion

In conclusion, the correct factorization of the trinomial $-3x^2 - 10x - 8$ is $-(x+4)(x+2)$ . This is because the signs of the two binomials must be different, and the factors of the constant term that add up to the coefficient of the $x$ term are $-8$ and $-2$ .

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Frequently Asked Questions

Q: What is a trinomial?

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

Q: How do I factor a trinomial?

To factor a trinomial, you need to follow these steps:

Identify the coefficients: Identify the coefficients of the trinomial, which are the numbers in front of the variable terms.
Determine the signs: Determine the signs of the coefficients. If the coefficient of the $x^2$ term is positive, the signs of the two binomials must be the same. If the coefficient of the $x^2$ term is negative, the signs of the two binomials must be different.
Find the factors: Find the factors of the constant term that add up to the coefficient of the $x$ term.
Write the factored form: Write the factored form of the trinomial by multiplying the two binomials.

Q: What are the common mistakes to avoid when factoring trinomials?

Some common mistakes to avoid when factoring trinomials include:

Not checking the signs: Make sure to check the signs of the coefficients and determine the correct signs for the two binomials.
Not finding the correct factors: Make sure to find the correct factors of the constant term that add up to the coefficient of the $x$ term.
Not writing the factored form correctly: Make sure to write the factored form of the trinomial by multiplying the two binomials correctly.

Q: Can I use a calculator to factor trinomials?

Yes, you can use a calculator to factor trinomials. However, it's always a good idea to check your work by factoring the trinomial manually to ensure that you get the correct answer.

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

A trinomial can be factored if it can be written as a product of two binomials. To determine if a trinomial can be factored, try to find two binomials whose product equals the original trinomial.

Q: What are some examples of trinomials that can be factored?

Some examples of trinomials that can be factored include:

$x^2 + 5x + 6$
$x^2 - 7x - 18$
$x^2 + 2x - 15$

Q: What are some examples of trinomials that cannot be factored?

Some examples of trinomials that cannot be factored include:

$x^2 + 2x + 1$
$x^2 - 4x + 4$
$x^2 + 3x + 2$

Conclusion

In conclusion, factoring trinomials can be a challenging task, but with practice and patience, you can master it. Remember to follow the steps outlined above, and don't be afraid to ask for help if you need it. With this guide, you'll be able to factor trinomials like a pro in no time!