Problem 83 - Elementary Algebra

Question

For the following problems, factor, if possible, the trinomials. $$ 36 a^{2}+60 a b+25 b^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial \(36a^2 + 60ab + 25b^2\) is \((6a + 5b)^2\).

Step 1: Identify the terms and their squares

In the given trinomial, we have - First term: \(36a^2\), which is a perfect square, as it can be written as \((6a)^2\). - Second term: \(60ab\), we need to check if it can be represented as a product of the square roots of the first and third term with a constant. - Third term: \(25b^2\), which is a perfect square, as it can be written as \((5b)^2\).

Step 2: Check if the trinomial is a perfect square trinomial

A quadratic trinomial is a perfect square trinomial if the middle term can be represented as the product of the square roots of the first and third terms multiplied by 2. In this case, $$ 2(6a)(5b) = 60ab $$ Since the middle term (second term) is equal to the product of the square roots of the first and third term multiplied by 2, we can conclude that the given trinomial is a perfect square trinomial.

Step 3: Factor the trinomial using the square of a binomial formula

Now we know that the trinomial is a perfect square trinomial, we can factor it as: $$ (A + B)^2 = A^2 + 2AB + B^2 $$ Here, - \(A = 6a\) - \(B = 5b\) So, the factored form of the trinomial is: $$ (6a + 5b)^2 $$ Therefore, the factored form of the given trinomial is: $$ 36a^2 + 60ab + 25b^2 = (6a + 5b)^2 $$

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic trinomial. It takes the form of \(A^2 + 2AB + B^2\) or \(A^2 - 2AB + B^2\), where \(A\) and \(B\) represent any algebraic expressions. To identify a perfect square trinomial, each term in the trinomial must be a perfect square, and the middle term must be twice the product of the square roots of the first and last terms.
In our exercise, the trinomial \(36a^2+60ab+25b^2\) fits this pattern perfectly, indicating that it's a perfect square trinomial. Recognizing this pattern is essential as it directs us to use the square of a binomial formula to factor the trinomial efficiently.

Quadratic Trinomial

A quadratic trinomial is an algebraic expression with three terms of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). To solve a quadratic trinomial, we typically look for two binomials that multiply together to give the original trinomial. This can be approached by various methods, such as factoring by grouping, the quadratic formula, or, in special cases, identifying it as a perfect square trinomial. The key to solving these kinds of problems lies in understanding the relationships between the coefficients and the terms.

Square of a Binomial Formula

The square of a binomial formula is a powerful algebraic identity that simplifies the process of squaring binomials. It states that \( (A + B)^2 = A^2 + 2AB + B^2 \) and similarly, \( (A - B)^2 = A^2 - 2AB + B^2 \). It's essential when factoring perfect square trinomials, as demonstrated in the solution to the given exercise.
By recognizing that the given trinomial is of the perfect square variety, we can apply this formula in reverse to factor it into \( (6a + 5b)^2 \), considerably simplifying the expression and revealing its underlying structure.

Algebraic Factoring

Algebraic factoring is the process of breaking down complex algebraic expressions into simpler, multiplicative components. This concept is fundamental in solving equations, simplifying expressions, and finding zeros or roots of functions. There are different techniques of factoring, including taking out the greatest common factor, factoring by grouping, and special patterns like the difference of squares and perfect square trinomials.
Understanding the different methods is crucial for students, as it allows them to tackle a wide variety of algebraic expressions. Practice is key when it comes to factoring, as it helps to quickly recognize patterns and determine the appropriate strategy.