Splitting the Middle Term

Definition

Splitting the middle term is a technique used in the factorization of trinomials of the form $ax^2 + bx + c$. It involves breaking down the middle term, $bx$, into two terms that, when multiplied, result in the original middle term.

5 Must Know Facts For Your Next Test

1. Splitting the middle term is a crucial step in factoring trinomials of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are integers.
2. The goal of splitting the middle term is to find two numbers that, when multiplied, give the original middle term $bx$, and when added, give the original middle term $bx$.
3. Splitting the middle term is particularly useful when the middle term $bx$ cannot be easily factored, and the trinomial cannot be easily recognized as a perfect square trinomial.
4. The process of splitting the middle term involves finding two numbers that, when multiplied, give the product $ac$, and when added, give the coefficient $b$.
5. Successful splitting of the middle term often leads to the factorization of the trinomial into the product of two binomials.

Review Questions

• Explain the purpose of splitting the middle term in the factorization of trinomials of the form $ax^2 + bx + c$.
  • The purpose of splitting the middle term $bx$ is to find two numbers that, when multiplied, give the product $ac$, and when added, give the coefficient $b$. This step is crucial because it allows the trinomial to be factored into the product of two binomials, which is the desired form for the factorization of such trinomials. By splitting the middle term, the trinomial can be expressed as the product of two simpler polynomial expressions, making it easier to work with and understand.
• Describe the process of splitting the middle term in the factorization of a trinomial of the form $ax^2 + bx + c$.
  • The process of splitting the middle term involves finding two numbers that, when multiplied, give the product $ac$, and when added, give the coefficient $b$. This is done by considering all possible pairs of numbers that, when multiplied, give $ac$, and then selecting the pair that, when added, results in $b$. Once the appropriate pair of numbers is found, the trinomial can be rewritten as the product of two binomials, which is the desired form for the factorization.
• Explain how the successful splitting of the middle term leads to the factorization of a trinomial of the form $ax^2 + bx + c$.
  • When the middle term $bx$ is successfully split into two terms that, when multiplied, give the product $ac$, and when added, give the coefficient $b$, the trinomial can be factored into the product of two binomials. This is because the trinomial can now be rewritten as the product of two simpler polynomial expressions, which is the goal of the factorization process. The ability to split the middle term is a crucial step in the factorization of trinomials, as it allows the trinomial to be expressed in a form that can be easily recognized and manipulated to find its factors.

Related terms

Trinomial: A polynomial expression with three terms, typically in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Factorization: The process of expressing a polynomial as a product of simpler polynomial factors.

Perfect Square Trinomial: A trinomial that can be expressed as the square of a binomial, in the form $(x + y)^2$.