ATHS Factoring General Trinomial

Summary

This educational video script explains the process of factoring quadratic trinomials using the 'ac method' with two cases. It demonstrates how to factor expressions like x^2 + 5x + 6 and x^2 - 7x + 12 by identifying pairs of factors that sum to the middle coefficient. The script also covers factoring general trinomials, such as 3x^2 + 13x + 4 and 4x^2 - 16x + 7, by pairing factors of the leading coefficient with those of the constant term, ensuring the inner and outer product sums match the middle term. Each example is methodically worked through to showcase the factoring technique.

Takeaways

• 🔢 The fourth type of factoring is called general factoring, which has two cases.
• 📝 The first case involves factoring trinomials like x² + 5x + 6. Start by identifying the factors of the constant (6) and find pairs that add up to the middle coefficient (5).
• 🧮 For x² + 5x + 6, the correct factor pair is (x + 3)(x + 2).
• 📘 Another example is x² - 7x + 12. The correct factor pair is (x - 3)(x - 4).
• 🔍 In the case of x² + 2x - 15, the correct factor pair is (x - 3)(x + 5).
• 🛠️ The second case of general factoring involves more complex trinomials, such as 3x² + 13x + 4. Start by identifying factors of the first and last terms.
• 📊 Position the factors in binomials, and then multiply the inner and outer terms to match the middle coefficient.
• ✔️ For 3x² + 13x + 4, the correct factor pair is (3x + 1)(x + 4).
• 🔗 Another example, 2x² - 3x - 9, factors to (2x + 3)(x - 3).
• 🔧 Finally, 4x² - 16x + 7 factors to (2x - 7)(2x - 1).

Q & A

What is the fourth type of factoring mentioned in the script?

The fourth type of factoring mentioned in the script is called 'general trinomials'.

How many cases are there for the general trinomial factoring method?

There are two cases for the general trinomial factoring method.

What is the first step in factoring a trinomial of the form x^2 + bx + c?

The first step in factoring a trinomial of the form x^2 + bx + c is to identify the factors of the constant term (c).

Why is it necessary to add the pairs of factors in the factoring process?

The pairs of factors are added to ensure that the sum of the products matches the middle coefficient (b) of the trinomial.

What is the correct pair of factors for the trinomial x^2 + 5x + 6?

The correct pair of factors for the trinomial x^2 + 5x + 6 is (x + 3) and (x + 2).

What is the sign of the factors when all the coefficients of the trinomial are positive?

When all the coefficients of the trinomial are positive, the factors will have the same signs.

How do you determine the correct pair of factors when there are multiple possibilities?

You determine the correct pair of factors by multiplying the inner and outer terms of each pair and checking if the sum matches the middle term of the trinomial.

What is the factored form of the trinomial 3x^2 + 13x + 4?

The factored form of the trinomial 3x^2 + 13x + 4 is (3x + 1)(x + 4).

What is the strategy for factoring a trinomial when the leading coefficient is not 1?

When the leading coefficient is not 1, you first factor out the greatest common factor and then apply the general trinomial factoring method to the remaining quadratic expression.

How does the script handle negative constants in the trinomial factoring process?

The script handles negative constants by considering pairs of factors that, when added, result in the negative middle term of the trinomial.

What is the significance of the middle term in the factoring process?

The middle term in the factoring process is significant because it determines the correct pair of factors when checking the sum of the products of the inner and outer terms.