Perfect square trinomials are algebraic expressions with three terms that are obtained by multiplying a binomial with the same binomial. A perfect square is a number that is obtained by multiplying a number by itself. Binomials are algebraic expressions consisting of just two terms that are either separated by a positive (+) or a negative (-) sign. Similarly, trinomials are algebraic expressions consisting of three terms. When a binomial consisting of a variable and a constant is multiplied by itself, it results in a perfect square trinomial having three terms. The terms of a perfect square trinomial are separated by either a positive or a negative sign.
1. Perfect Square Trinomial Definition 2. Perfect Square Trinomial Pattern 3. How to Factor a Perfect Square Trinomial? 4. Perfect Square Trinomial Formula 5. Perfect Square Trinomial FAQs
A perfect square trinomial is defined as an algebraic expression that is obtained by squaring a binomial expression. It is of the form ax2 + bx + c. Here a, b, and c are real numbers and a ≠ 0. For example, let us take a binomial (x+4) and multiply it to (x+4). The result obtained is x2 + 8x + 16. A perfect square trinomial can be decomposed into two binomials and the binomials when multiplied with each other gives the perfect square trinomial.
Perfect square trinomial generally has the pattern of a2 + 2ab + b2or a2 – 2ab + b2. Given a binomial, to find the perfect square trinomial, we follow the steps given below. They are,
- Step 1: Find the square the first term of the binomial.
- Step 2: Multiply the first and the second term of the binomial with 2.
- Step 3: Find the square of the second term of the binomial.
- Step 4: Sum up all the three terms obtained in steps 1, 2, and 3.
The first term of the perfect square trinomial is the square of the first term of the binomial. and the second term is twice the product of the two terms of the binomial and the third term is the square of the second term of the binomial. Take a look at the figure shown below to understand the perfect square trinomial pattern. If the binomial being squared has a positive sign, then all the terms in the perfect square trinomial are positive, whereas, if the binomial has a negative sign attached with its second term, then the second term of the trinomial (which is twice the product of the two variables) will be negative.
Perfect square trinomials are either separated by a positive or a negative symbol between the terms. Two important algebraic identities with regards to perfect square trinomial are as follows.
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
The steps to be followed to factor a perfect square polynomial are as follows.
- Write the given perfect square trinomial of the form a2 + 2ab + b2 or a2 – 2ab + b2, such that the first and the third terms are perfect squares, one being a variable and another being a constant.
- Check if the middle term is twice the product of the first and the third term. Also, check the sign of the middle term.
- If the middle term is positive, then compare the perfect square trinomial with a2 + 2ab + b2 and if the middle term is negative, then compare the perfect square trinomial with a2 – 2ab + b2.
- If the middle term is positive, then the factors are (a+b) (a+b) and if the middle term is negative, then the factors are (a-b) (a-b).
Perfect square trinomial is obtained by multiplying the same binomial expression with each other. A trinomial is said to be a perfect square if it is of the form ax2+bx+c and also satisfies the condition of b2 = 4ac. There are two forms of a perfect square trinomial. They are,
- (ax)2+ 2abx + b2= (ax + b)2- (1)
- (ax)2−2abx + b2 = (ax−b)2- (2)
For example, let us take a perfect square polynomial, x2 + 6x + 9. Comparing this with the form ax2+bx+c, we get a = 1, b = 6 and c = 9. Let us check if this trinomial satisfies the condition b2 = 4ac. b2 = 36 and 4 × a × c = 4 × 1 × 9, which is equal to 36. Therefore, the trinomial satisfies the condition b2 = 4ac. So, we can call it a perfect square trinomial.
Topics Related to Perfect Square Trinomial
Check out some important topics related to perfect square trinomial.
- Algebra
- Factorization of Algebraic Expressions
- Squaring a Trinomial
- Perfect Square Trinomial Formula