Factor The Trinomial.$\[ A^2 + 6a - 16 \\]

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What is a Trinomial?

A trinomial is a polynomial expression consisting of three terms. It is a quadratic expression that can be factored into the product of two binomials. Factoring trinomials is an essential skill in algebra, as it allows us to simplify complex expressions and solve equations.

The General Form of a Trinomial

The general form of a trinomial is:

ax2+bx+c{ ax^2 + bx + c }

where a, b, and c are constants, and x is the variable. In the given trinomial, a2+6aβˆ’16a^2 + 6a - 16, we can identify the values of a, b, and c as follows:

  • a = 1
  • b = 6
  • c = -16

Factoring Trinomials: A Step-by-Step Guide

To factor a trinomial, we need to find two binomials whose product is equal to the original trinomial. The general form of a binomial is:

(x+m)(x+n){ (x + m)(x + n) }

where m and n are constants. We can use the following steps to factor a trinomial:

Step 1: Identify the Values of a, b, and c

In the given trinomial, a2+6aβˆ’16a^2 + 6a - 16, we have already identified the values of a, b, and c as follows:

  • a = 1
  • b = 6
  • c = -16

Step 2: Determine the Signs of the Binomials

To determine the signs of the binomials, we need to examine the signs of the terms in the trinomial. If the middle term is positive, the binomials will have the same sign. If the middle term is negative, the binomials will have opposite signs.

In the given trinomial, the middle term is positive (6a), so the binomials will have the same sign.

Step 3: Find the Factors of the Constant Term

The constant term is -16. We need to find two numbers whose product is equal to -16 and whose sum is equal to the coefficient of the middle term (6).

The factors of -16 are:

  • 1 and -16
  • 2 and -8
  • 4 and -4

We can see that the sum of 2 and -8 is equal to 6, so we can use these numbers to form the binomials.

Step 4: Write the Binomials

Using the numbers 2 and -8, we can write the binomials as follows:

(x+2)(xβˆ’8){ (x + 2)(x - 8) }

Step 5: Check the Factored Form

To check the factored form, we can multiply the binomials together to get the original trinomial.

(x+2)(xβˆ’8)=x2βˆ’8x+2xβˆ’16{ (x + 2)(x - 8) = x^2 - 8x + 2x - 16 }

=x2βˆ’6xβˆ’16{ = x^2 - 6x - 16 }

We can see that the factored form is not equal to the original trinomial. This is because we made an error in the previous steps.

Step 6: Correct the Error

Let's go back to Step 3 and try again. We need to find two numbers whose product is equal to -16 and whose sum is equal to the coefficient of the middle term (6).

The factors of -16 are:

  • 1 and -16
  • 2 and -8
  • 4 and -4

We can see that the sum of 4 and -4 is equal to 0, not 6. However, we can see that the sum of 2 and -8 is equal to 0, not 6. But we can see that the sum of 1 and -16 is equal to -15, not 6. However, we can see that the sum of 8 and -8 is equal to 0, not 6. But we can see that the sum of 4 and -12 is equal to -8, not 6. However, we can see that the sum of 6 and -10 is equal to -4, not 6. But we can see that the sum of 7 and -9 is equal to -2, not 6. However, we can see that the sum of 8 and -8 is equal to 0, not 6. But we can see that the sum of 9 and -7 is equal to 2, not 6. However, we can see that the sum of 10 and -6 is equal to 4, not 6. However, we can see that the sum of 11 and -5 is equal to 6, so we can use these numbers to form the binomials.

Step 7: Write the Binomials

Using the numbers 11 and -5, we can write the binomials as follows:

(x+11)(xβˆ’5){ (x + 11)(x - 5) }

Step 8: Check the Factored Form

To check the factored form, we can multiply the binomials together to get the original trinomial.

(x+11)(xβˆ’5)=x2βˆ’5x+11xβˆ’55{ (x + 11)(x - 5) = x^2 - 5x + 11x - 55 }

=x2+6xβˆ’55{ = x^2 + 6x - 55 }

We can see that the factored form is not equal to the original trinomial. This is because we made an error in the previous steps.

Step 9: Correct the Error

Let's go back to Step 3 and try again. We need to find two numbers whose product is equal to -16 and whose sum is equal to the coefficient of the middle term (6).

The factors of -16 are:

  • 1 and -16
  • 2 and -8
  • 4 and -4

We can see that the sum of 2 and -8 is equal to 0, not 6. However, we can see that the sum of 1 and -16 is equal to -15, not 6. However, we can see that the sum of 8 and -8 is equal to 0, not 6. But we can see that the sum of 4 and -12 is equal to -8, not 6. However, we can see that the sum of 6 and -10 is equal to -4, not 6. But we can see that the sum of 7 and -9 is equal to -2, not 6. However, we can see that the sum of 8 and -8 is equal to 0, not 6. But we can see that the sum of 9 and -7 is equal to 2, not 6. However, we can see that the sum of 10 and -6 is equal to 4, not 6. However, we can see that the sum of 11 and -5 is equal to 6, so we can use these numbers to form the binomials.

However, we can see that the product of 11 and -5 is equal to -55, not -16. However, we can see that the product of 8 and -2 is equal to -16, and the sum of 8 and -2 is equal to 6, so we can use these numbers to form the binomials.

Step 10: Write the Binomials

Using the numbers 8 and -2, we can write the binomials as follows:

(x+8)(xβˆ’2){ (x + 8)(x - 2) }

Step 11: Check the Factored Form

To check the factored form, we can multiply the binomials together to get the original trinomial.

(x+8)(xβˆ’2)=x2βˆ’2x+8xβˆ’16{ (x + 8)(x - 2) = x^2 - 2x + 8x - 16 }

=x2+6xβˆ’16{ = x^2 + 6x - 16 }

We can see that the factored form is equal to the original trinomial.

Conclusion

Q: What is a trinomial?

A: A trinomial is a polynomial expression consisting of three terms. It is a quadratic expression that can be factored into the product of two binomials.

Q: What is the general form of a trinomial?

A: The general form of a trinomial is:

ax2+bx+c{ 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 general form of a binomial is:

(x+m)(x+n){ (x + m)(x + n) }

where m and n are constants.

Q: What are the steps to factor a trinomial?

A: The steps to factor a trinomial are:

  1. Identify the values of a, b, and c.
  2. Determine the signs of the binomials.
  3. Find the factors of the constant term.
  4. Write the binomials.
  5. Check the factored form.

Q: What if I make a mistake while factoring a trinomial?

A: If you make a mistake while factoring a trinomial, you can go back to the previous steps and try again. You can also use the factored form to check your work.

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

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

Q: Can I factor a trinomial with a zero middle term?

A: Yes, you can factor a trinomial with a zero middle term. The process is the same as factoring a trinomial with a non-zero middle term.

Q: Can I factor a trinomial with a negative constant term?

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

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

A: You can check if a trinomial can be factored by looking at the constant term. If the constant term is a perfect square, then the trinomial can be factored.

Q: Can I factor a trinomial with a variable in the constant term?

A: No, you cannot factor a trinomial with a variable in the constant term. The constant term must be a constant.

Q: Can I factor a trinomial with a fraction in the constant term?

A: No, you cannot factor a trinomial with a fraction in the constant term. The constant term must be a constant.

Q: Can I factor a trinomial with a negative variable in the constant term?

A: No, you cannot factor a trinomial with a negative variable in the constant term. The constant term must be a constant.

Q: Can I factor a trinomial with a variable in the middle term?

A: Yes, you can factor a trinomial with a variable in the middle term. The process is the same as factoring a trinomial with a constant in the middle term.

Q: Can I factor a trinomial with a fraction in the middle term?

A: No, you cannot factor a trinomial with a fraction in the middle term. The middle term must be a constant.

Q: Can I factor a trinomial with a negative variable in the middle term?

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

Q: Can I factor a trinomial with a variable in the leading coefficient?

A: No, you cannot factor a trinomial with a variable in the leading coefficient. The leading coefficient must be a constant.

Q: Can I factor a trinomial with a fraction in the leading coefficient?

A: No, you cannot factor a trinomial with a fraction in the leading coefficient. The leading coefficient must be a constant.

Q: Can I factor a trinomial with a negative variable in the leading coefficient?

A: No, you cannot factor a trinomial with a negative variable in the leading coefficient. The leading coefficient must be a constant.

Conclusion

Factoring trinomials is an essential skill in algebra. By following the steps outlined in this guide, you can factor trinomials into the product of two binomials. Remember to check your work and use the factored form to verify your answer.