📚 Topic Summary
Factoring trinomials of the form $ax^2 + bx + c$ using the grouping method involves rewriting the middle term ($bx$) as a sum of two terms whose coefficients multiply to $ac$ and add up to $b$. Once you rewrite the middle term, you can factor by grouping the first two terms and the last two terms. This process simplifies the trinomial into a product of two binomials. Let's practice!
For example, to factor $2x^2 + 7x + 3$, we need to find two numbers that multiply to $2*3 = 6$ and add up to $7$. Those numbers are $6$ and $1$. So, we rewrite the trinomial as $2x^2 + 6x + x + 3$, then factor by grouping: $2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$.
🔤 Part A: Vocabulary
| Term |
Definition |
| Trinomial |
A. A polynomial with four terms. |
| Factor |
B. To express a number or algebraic expression as a product of two or more factors. |
| Coefficient |
C. A polynomial with three terms. |
| Grouping |
D. The numerical or constant quantity placed before and multiplying the variable in an algebraic expression. |
| Polynomial |
E. A method of factoring by pairing terms that have a common monomial factor. |
Match the term to the correct definition.
✍️ Part B: Fill in the Blanks
To factor a trinomial $ax^2 + bx + c$ by grouping, we first find two numbers that multiply to _____ and add up to _____. We then rewrite the middle term, $bx$, using these two numbers. After rewriting, we factor by _____ the first two terms and the last two terms. This results in two binomials that share a common _____. Factoring out the common binomial gives us the factored form of the original trinomial.
🤔 Part C: Critical Thinking
Explain in your own words why the grouping method works for factoring trinomials. Provide an example to illustrate your explanation.