Factoring the Trinomial using the Quadratic Formula

Oct 13, 2025 by windailys.com 24 views

Which of the Binomials Below is a Factor of this Trinomial?

Understanding the Problem

When we are given a trinomial and asked to find its factors, we need to understand that a trinomial can be factored into the product of two binomials. In this problem, we are given a trinomial in the form of 2x^2 - 3x - 10 and asked to determine which of the given binomials is a factor of this trinomial.

The Trinomial: 2x^2 - 3x - 10

To find the factors of the trinomial, we need to first identify the coefficients of the terms. In this case, the coefficients are 2, -3, and -10. We also need to identify the variables, which in this case is x.

The Binomials

We are given four binomials to choose from: A. x - 1, B. x + 2, C. x - 2, and D. x + 1. We need to determine which of these binomials is a factor of the trinomial.

Factoring the Trinomial

To factor the trinomial, we need to find two binomials whose product is equal to the trinomial. We can start by looking for two numbers whose product is equal to the product of the coefficients of the trinomial (2 * -10 = -20) and whose sum is equal to the coefficient of the middle term (-3).

Using the Factor Theorem

The factor theorem states that if a polynomial f(x) is divided by (x - a), then the remainder is f(a). We can use this theorem to test each of the binomials and see if it is a factor of the trinomial.

Testing Binomial A: x - 1

To test binomial A, we need to substitute x - 1 into the trinomial and see if the remainder is zero.

2(x - 1)^2 - 3(x - 1) - 10

Expanding the expression, we get:

2(x^2 - 2x + 1) - 3x + 3 - 10

Simplifying the expression, we get:

2x^2 - 4x + 2 - 3x + 3 - 10

Combine like terms:

2x^2 - 7x - 5

The remainder is not zero, so binomial A is not a factor of the trinomial.

Testing Binomial B: x + 2

To test binomial B, we need to substitute x + 2 into the trinomial and see if the remainder is zero.

2(x + 2)^2 - 3(x + 2) - 10

Expanding the expression, we get:

2(x^2 + 4x + 4) - 3x - 6 - 10

Simplifying the expression, we get:

2x^2 + 8x + 8 - 3x - 6 - 10

Combine like terms:

2x^2 + 5x - 8

The remainder is not zero, so binomial B is not a factor of the trinomial.

Testing Binomial C: x - 2

To test binomial C, we need to substitute x - 2 into the trinomial and see if the remainder is zero.

2(x - 2)^2 - 3(x - 2) - 10

Expanding the expression, we get:

2(x^2 - 4x + 4) - 3x + 6 - 10

Simplifying the expression, we get:

2x^2 - 8x + 8 - 3x + 6 - 10

Combine like terms:

2x^2 - 11x + 4

The remainder is not zero, so binomial C is not a factor of the trinomial.

Testing Binomial D: x + 1

To test binomial D, we need to substitute x + 1 into the trinomial and see if the remainder is zero.

2(x + 1)^2 - 3(x + 1) - 10

Expanding the expression, we get:

2(x^2 + 2x + 1) - 3x - 3 - 10

Simplifying the expression, we get:

2x^2 + 4x + 2 - 3x - 3 - 10

Combine like terms:

2x^2 + x - 11

The remainder is not zero, so binomial D is not a factor of the trinomial.

Conclusion

After testing each of the binomials, we can conclude that none of them are factors of the trinomial 2x^2 - 3x - 10. However, we can try to factor the trinomial using other methods.

Factoring the Trinomial using the Quadratic Formula

The quadratic formula states that the solutions to the quadratic equation ax^2 + bx + c = 0 are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

We can use this formula to find the solutions to the trinomial 2x^2 - 3x - 10.

Solving the Trinomial using the Quadratic Formula

To solve the trinomial, we need to plug in the values of a, b, and c into the quadratic formula.

a = 2 b = -3 c = -10

x = (3 ± √((-3)^2 - 4(2)(-10))) / 2(2)

x = (3 ± √(9 + 80)) / 4

x = (3 ± √89) / 4

The solutions to the trinomial are x = (3 + √89) / 4 and x = (3 - √89) / 4.

Conclusion

In conclusion, we have found that none of the given binomials are factors of the trinomial 2x^2 - 3x - 10. However, we have also found the solutions to the trinomial using the quadratic formula.
Q&A: Which of the Binomials Below is a Factor of this Trinomial?

Q: What is the main goal of factoring a trinomial?

A: The main goal of factoring a trinomial is to express it as the product of two binomials. This can help us solve equations and find the roots of the trinomial.

Q: How do we determine which binomial is a factor of a trinomial?

A: To determine which binomial is a factor of a trinomial, we need to test each binomial by substituting it into the trinomial and seeing if the remainder is zero. If the remainder is zero, then the binomial is a factor of the trinomial.

Q: What is the factor theorem?

A: The factor theorem states that if a polynomial f(x) is divided by (x - a), then the remainder is f(a). We can use this theorem to test each binomial and see if it is a factor of the trinomial.

Q: How do we use the quadratic formula to solve a trinomial?

A: To use the quadratic formula to solve a trinomial, we need to plug in the values of a, b, and c into the formula. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: What are the solutions to the trinomial 2x^2 - 3x - 10?

A: The solutions to the trinomial 2x^2 - 3x - 10 are x = (3 + √89) / 4 and x = (3 - √89) / 4.

Q: Why is it important to check if a binomial is a factor of a trinomial?

A: It is important to check if a binomial is a factor of a trinomial because it can help us solve equations and find the roots of the trinomial. If a binomial is a factor of a trinomial, then we can use it to find the solutions to the trinomial.

Q: Can a trinomial have more than one factor?

A: Yes, a trinomial can have more than one factor. In fact, a trinomial can have multiple factors, and we need to test each binomial to see if it is a factor of the trinomial.

Q: How do we know if a binomial is a factor of a trinomial?

A: We know if a binomial is a factor of a trinomial by testing it using the factor theorem. If the remainder is zero, then the binomial is a factor of the trinomial.

Q: Can we use the quadratic formula to solve any trinomial?

A: Yes, we can use the quadratic formula to solve any trinomial. However, we need to make sure that the trinomial is in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: What is the significance of the solutions to a trinomial?

A: The solutions to a trinomial are the values of x that make the trinomial equal to zero. These values are important because they can help us solve equations and find the roots of the trinomial.

Q: Can we use the solutions to a trinomial to find the roots of the trinomial?

A: Yes, we can use the solutions to a trinomial to find the roots of the trinomial. The roots of a trinomial are the values of x that make the trinomial equal to zero.

Q: How do we use the solutions to a trinomial to find the roots of the trinomial?

A: We can use the solutions to a trinomial to find the roots of the trinomial by substituting the solutions into the trinomial and solving for x.