Dividing Trinomials by Binomials: Long Division vs. Factoring for GED Math

Factoring a Trinomial by a Binomial

• 💡 To factor a trinomial with a leading coefficient of one, focus on the constant term.
• 🎯 Find two numbers that multiply to the constant term and add to the coefficient of the middle term.
• 🔑 For example, to factor x² + 3x - 28, find numbers that multiply to -28 and add to 3. These are -4 and 7.
• ✅ The factored form is (x - 4)(x + 7).

Simplifying the Expression

• 🚀 When dividing (x - 4)(x + 7) by (x - 4), the (x - 4) terms cancel out.
• ✨ The simplified result is x + 7.

Long Division Method

• 🔬 Set up the long division with the binomial (x - 4) as the divisor and the trinomial (x² + 3x - 28) as the dividend.
• 🧠 Divide the first term of the dividend (x²) by the first term of the divisor (x) to get x.
• ➕ Multiply x by the divisor (x - 4) to get x² - 4x, then subtract this from the dividend.
• ➡️ Bring down the next term (-28) and repeat the process: divide the new first term (7x) by x to get 7.
• ➖ Multiply 7 by the divisor (x - 4) to get 7x - 28, then subtract this from the remaining terms.
• 💯 A remainder of zero indicates the division is complete, and the quotient is x + 7.

Conclusion

• 🤝 Both factoring and long division yield the same result when dividing a trinomial by a binomial.
• 💡 Factoring is often a simpler and quicker method when applicable.