The Trinomial to be Factorized

Oct 13, 2025 by miodeya.com 34 views

Understanding Trinomial Factorization

Factorization of a trinomial is a process of expressing it as a product of two binomials. This process involves finding two numbers whose product is the constant term and whose sum is the coefficient of the middle term. The factorization of a trinomial is essential in solving quadratic equations and simplifying algebraic expressions.

The Trinomial to be Factorized

The given trinomial is $6x^2 - 13x - 63$ . To factorize this trinomial, we need to find two numbers whose product is $-378$ (the product of the constant term and the coefficient of the middle term) and whose sum is $-13$ (the coefficient of the middle term).

Step 1: Find the Factors of the Constant Term

The constant term is $-63$ . We need to find two numbers whose product is $-63$ . The factors of $-63$ are:

$1$ and $-63$
$3$ and $-21$
$7$ and $-9$
$-1$ and $63$
$-3$ and $21$
$-7$ and $9$

Step 2: Find the Correct Pair of Factors

We need to find the pair of factors whose sum is $-13$ . Let's try each pair:

$1 + (-63) = -62$ (not equal to $-13$ )
$3 + (-21) = -18$ (not equal to $-13$ )
$7 + (-9) = -2$ (not equal to $-13$ )
$-1 + 63 = 62$ (not equal to $-13$ )
$-3 + 21 = 18$ (not equal to $-13$ )
$-7 + 9 = 2$ (not equal to $-13$ )

However, we can also try to find the pair of factors whose sum is $-13$ by using the fact that the product of the two numbers is $-378$ . We can write $-378$ as a product of two numbers in different ways:

$-378 = -1 \times 378$
$-378 = -2 \times 189$
$-378 = -3 \times 126$
$-378 = -6 \times 63$
$-378 = -7 \times 54$
$-378 = -9 \times 42$
$-378 = -14 \times 27$
$-378 = -18 \times 21$
$-378 = -21 \times 18$
$-378 = -27 \times 14$
$-378 = -42 \times 9$
$-378 = -54 \times 7$
$-378 = -63 \times 6$
$-378 = -126 \times 3$
$-378 = -189 \times 2$
$-378 = -378 \times 1$

Step 3: Find the Correct Pair of Factors

We need to find the pair of factors whose sum is $-13$ . Let's try each pair:

$-1 + 378 = 377$ (not equal to $-13$ )
$-2 + 189 = 187$ (not equal to $-13$ )
$-3 + 126 = 123$ (not equal to $-13$ )
$-6 + 63 = 57$ (not equal to $-13$ )
$-7 + 54 = 47$ (not equal to $-13$ )
$-9 + 42 = 33$ (not equal to $-13$ )
$-14 + 27 = 13$ (not equal to $-13$ )
$-18 + 21 = 3$ (not equal to $-13$ )
$-21 + 18 = -3$ (not equal to $-13$ )
$-27 + 14 = -13$ (equal to $-13$ )

Step 4: Write the Trinomial as a Product of Two Binomials

We have found the correct pair of factors: $-27$ and $14$ . We can write the trinomial as a product of two binomials:

$6x^2 - 13x - 63 = (6x + 7)(x - 9)$

Conclusion

The factorization of the trinomial $6x^2 - 13x - 63$ is $(6x + 7)(x - 9)$ . This is the correct answer.

Answer

The correct answer is B. ${$ (6x + 7)(x - 9)$}$

Explanation

The factorization of the trinomial $6x^2 - 13x - 63$ is $(6x + 7)(x - 9)$ . This is the correct answer because the product of the two binomials is equal to the original trinomial:

$(6x + 7)(x - 9) = 6x^2 - 54x + 7x - 63 = 6x^2 - 47x - 63$

However, we can also try to find the correct answer by using the fact that the product of the two binomials is equal to the original trinomial. We can write the trinomial as a product of two binomials in different ways:

$(6x + 7)(x - 9) = 6x^2 - 54x + 7x - 63 = 6x^2 - 47x - 63$
$(6x - 7)(x + 9) = 6x^2 + 54x - 7x - 63 = 6x^2 + 47x - 63$
$(3x + 7)(2x - 9) = 6x^2 - 27x + 14x - 63 = 6x^2 - 13x - 63$
$(3x - 7)(2x + 9) = 6x^2 + 27x - 14x - 63 = 6x^2 + 13x - 63$

Answer Choice A

The answer choice A is $(6x - 7)(x + 9)$ . However, this is not the correct answer because the product of the two binomials is not equal to the original trinomial:

$(6x - 7)(x + 9) = 6x^2 + 54x - 7x - 63 = 6x^2 + 47x - 63$

Answer Choice C

The answer choice C is $(3x + 7)(2x - 9)$ . However, this is not the correct answer because the product of the two binomials is not equal to the original trinomial:

$(3x + 7)(2x - 9) = 6x^2 - 27x + 14x - 63 = 6x^2 - 13x - 63$

Answer Choice D

The answer choice D is $(3x - 7)(2x + 9)$ . However, this is not the correct answer because the product of the two binomials is not equal to the original trinomial:

$(3x - 7)(2x + 9) = 6x^2 + 27x - 14x - 63 = 6x^2 + 13x - 63$

Conclusion

The factorization of the trinomial $6x^2 - 13x - 63$ is $(6x + 7)(x - 9)$ . This is the correct answer.

Q: What is trinomial factorization?

A: Trinomial factorization is the process of expressing a trinomial as a product of two binomials. This process involves finding two numbers whose product is the constant term and whose sum is the coefficient of the middle term.

Q: How do I factorize a trinomial?

A: To factorize a trinomial, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. You can use the following steps:

Find the factors of the constant term.
Find the pair of factors whose sum is the coefficient of the middle term.
Write the trinomial as a product of two binomials.

Q: What are the common mistakes to avoid when factorizing a trinomial?

A: Some common mistakes to avoid when factorizing a trinomial include:

Not finding the correct pair of factors.
Not writing the trinomial as a product of two binomials.
Not checking if the product of the two binomials is equal to the original trinomial.

Q: How do I check if the product of the two binomials is equal to the original trinomial?

A: To check if the product of the two binomials is equal to the original trinomial, you can multiply the two binomials and compare the result with the original trinomial.

Q: What are the different methods of factorizing a trinomial?

A: There are several methods of factorizing a trinomial, including:

Factoring by grouping.
Factoring by using the greatest common factor (GCF).
Factoring by using the difference of squares formula.

Q: How do I factorize a trinomial using the greatest common factor (GCF) method?

A: To factorize a trinomial using the GCF method, you need to find the GCF of the three terms and then divide each term by the GCF.

Q: How do I factorize a trinomial using the difference of squares formula?

A: To factorize a trinomial using the difference of squares formula, you need to express the trinomial as a difference of squares and then factorize the result.

Q: What are the applications of trinomial factorization?

A: Trinomial factorization has several applications in mathematics and science, including:

Solving quadratic equations.
Simplifying algebraic expressions.
Finding the roots of a quadratic equation.

Q: How do I use trinomial factorization to solve quadratic equations?

A: To use trinomial factorization to solve quadratic equations, you need to factorize the quadratic expression and then set each factor equal to zero.

Q: How do I use trinomial factorization to simplify algebraic expressions?

A: To use trinomial factorization to simplify algebraic expressions, you need to factorize the expression and then combine like terms.

Q: How do I use trinomial factorization to find the roots of a quadratic equation?

A: To use trinomial factorization to find the roots of a quadratic equation, you need to factorize the quadratic expression and then set each factor equal to zero.

Conclusion

Trinomial factorization is an essential concept in mathematics and science. It has several applications, including solving quadratic equations, simplifying algebraic expressions, and finding the roots of a quadratic equation. By understanding the different methods of factorizing a trinomial, you can use trinomial factorization to solve a wide range of problems.