Perfect Square Trinomial (PST) Information Card

1. What is a Perfect Square Trinomial?

A Perfect Square Trinomial (PST) is a special form of a quadratic trinomial that can be expressed as the square of a binomial. In general, it takes the form:

$ax^2 + bx + c$

where the trinomial is a perfect square if it can be factored into:

$(mx + n)^2$

Expanding this squared binomial gives:

$(mx + n)^2 = m^2x^2 + 2mnx + n^2$

So, a trinomial of the form $ax^2 + bx + c$ is a PST if:

• $a = m^2$
• $b = 2mn$
• $c = n^2$

2. When Do We Say That a Trinomial is a PST?

A trinomial $ax^2 + bx + c$ is called a Perfect Square Trinomial if it meets the following conditions:

1. Square Term: The first term $ax^2$ must be a perfect square.
2. Constant Term: The last term $c$ must be a perfect square.
3. Middle Term Relationship: The middle term $b$ should be twice the product of the square roots of the first and last terms.

Mathematically, if $a = m^2$ and $c = n^2$, then $b$ should equal $2mn$.

3. How to Factor a Perfect Square Trinomial?

To factor a Perfect Square Trinomial, follow these steps:

1. Identify the Square Roots: Determine the square roots of the first term $ax^2$ and the last term $c$.

2. Check the Middle Term: Ensure the middle term $b$ is twice the product of these square roots.

3. Write the Binomial: Express the trinomial as the square of a binomial: $ax^2 + bx + c = (mx + n)^2$ where $m$ is the square root of the first term and $n$ is the square root of the last term.

   Example: Factor the trinomial $x^2 + 6x + 9$.

   • The square root of $x^2$ is $x$.
   • The square root of $9$ is $3$.
   • The middle term $6x$ is twice the product of $x$ and $3$, confirming it's a PST.
   • Factor as $(x + 3)^2$.

Want more details or have any questions?

Related Questions

1. How can you identify a perfect square trinomial in a polynomial?
2. What are some examples of perfect square trinomials and their factorizations?
3. Can a perfect square trinomial have negative terms? How does that affect factoring?
4. How do perfect square trinomials relate to the difference of squares?
5. What is the difference between a perfect square trinomial and a general trinomial?

Tip:

When factoring, always check if the trinomial can be rewritten as a square of a binomial first; it simplifies the problem and reduces errors.