Mastering Perfect Square Trinomials: A Comprehensive Guide

Unlock the secrets of perfect square trinomials with our expert guide. Learn to identify, factor, and solve these special polynomials, enhancing your algebra prowess and problem-solving skills.

What is a perfect square trinomial

Let's first remember what a trinomial is. A polynomial has several terms. A trinomial (as the prefix "tri-" suggests) is a polynomial with three terms. When we're dealing with perfect squares, it means we're dealing with squaring binomials. Continue on to learn how we go about factoring a trinomial.

How to factor perfect square trinomials

One good way to recognize if a trinomial is perfect square is to look at its first and third term. If they are both squares, there's a good chance that you may be working with a perfect square trinomial.

Let's say we're working with the following: $x^{2}+14x+49$. Is this a perfect square trinomial? Looking at the first term, we've got $x^{2}$, which is a square. The last term is $49$. It is also a square since when you multiply $7$ by $7$, you'll get $49$. Therefore $49$ can also be written as $7^{2}$. The next step to identifying if we've got a perfect square is to see if we are able to get the middle term of $14x$ when we have $x^{2}$and $7^{2}$ to work with.

In the case of a perfect square, the middle term is the first term multiplied by the last term, and then multiplied by $2$. In other words, the perfect square trinomial formula is:

$a^{2} \pm ab + b^{2}$. We're now trying to see if we can get the middle term of $2ab$.

Since we've got our $a$ term as $x$, and our $b$ term as $7$, our $2ab$ becomes $2 \bullet 7 \bullet x$. That gives us a total of $14x$, which is the middle term in $x^{2}+14x+49$! Therefore, we can rewrite the question as $(x + 7)^{2}$through factoring perfect square trinomials. You've solved a perfect square trinomial! You're now ready to apply trinomial factoring to some practice problems.

Example problems

Question 1:

Factor the perfect square

$x^{2} - 2x + 36$

Solution:

We know that this is a perfect square, and all we're asked is to factor it. Therefore, just take a look at the first and last term and find what they are squares of. It'll give us:

$(x - 6)^{2}$

Question 2:

Factor the perfect square

$3x^{2} - 30x +75$

Solution:

Take out the common factor $3$

$3(x^{2} - 10x + 25)$

Factor the $x^{2} - 10x + 25$ and get the final answer:

$3(x - 5)^{2}$

Question 3:

Find the square of a binomial:

$(-3x^{2} + 3y^{2})^{2}$

Solution:

You can square it and it will become what we have here:

$ax^{2} - bxy +cy^{2}$

So the first term:

Square of $-3x^{2} = 9x^{4}$

The third term:

$3y^{2} = 9y^{4}$

The middle term is the multiplication of original $1^{st}$ and $2^{nd}$ term, and then times $2$

$-3x^{2} \bullet 3y^{2} = -9x^{2}y^{2}$

Then times $2$:

$-18x^{2}y^{2}$

So the final answer:

$(9x^{4} - 18x^{2}y^{2} + 9y^{4})$

To double check your answers, this online calculator will help you factor a polynomial expression. Use it as a reference, but make sure you learn how to properly go through the steps to answering a perfect square trinomial question.

Wasn't quite sure on the concepts covered in this chapter? Perhaps you may want to go back and review how to find common factors of polynomials or how to factor by grouping. Also read up on solving polynomials with unknown coefficients, and the intro to factoring polynomials.

Ready to move on? Up next, learn how to complete the square and change a quadratic function from standard form to vertex form.

• Common factors of polynomials
• Factoring polynomials by grouping
• Solving polynomials with unknown coefficients
• Factoring polynomials: x^2 + bx + c

• Completing the square
• Converting from general to vertex form by completing the square