Problem 14 - Elementary Algebra

Question

Factor, if possible, the following trinomials. $$ x^{2}+12 x+49 $$

Step-by-Step Solution

Answer

Question: Factor the trinomial $$x^{2}+12 x+49$$. Answer: The factored form of the given trinomial is $$(x + 7)^2$$.

Step 1: Analyze the given trinomial

Given the trinomial $$x^{2}+12 x+49$$, we notice that the constant term, 49, is a perfect square (49 = 7^2). Furthermore, the coefficient of the middle term, 12, is twice the square root of 49 (12 = 2 * 7).

Step 2: Factor the trinomial using the perfect square pattern

Since the given trinomial is in the form of $$x^2 + 2ax + a^2$$, where a=7, we can apply the perfect square pattern and rewrite it as a binomial square: $$ (x + a)^2 = (x + 7)^2 $$ So, the factored form of the given trinomial is $$(x + 7)^2$$.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It allows us to generalize arithmetical operations and to write and manipulate expressions in an abstract way.

In the context of factoring trinomials, algebra teaches us how to break down polynomial expressions into simpler products of factors. This is crucial for solving equations, simplifying expressions, and understanding the properties of graphs. Factoring is like a mathematical 'treasure hunt' where we find the numbers or expressions that multiply together to get the original polynomial.

For example, consider the expression \( x^2 + 12x + 36 \). Factoring in algebra would involve finding binomials, such as \( (x + 6) \), that when multiplied by itself, \( (x + 6)(x + 6) \), gives us back the original quadratic expression. This is a practical application of algebra, making complex problems more manageable by revealing their underlying structure.

Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression that is the result of squaring a binomial. It takes the generalized form of \( a^2 + 2ab + b^2 \), where \( a \) and \( b \) are real numbers or algebraic expressions. The key characteristics of a perfect square trinomial include a first term that is a perfect square, a last term that is also a perfect square, and a middle term that is twice the product of the square root of the first and last terms.

For instance, the expression \( x^2 + 12x + 36 \) is a perfect square trinomial because the first term, \( x^2 \), and the last term, \( 36 \), are perfect squares (since \( 36 = 6^2 \)), and the middle term, \( 12x \), is double the product of the square roots of the first and last terms (as \( 2 \times x \times 6 = 12x \)). Perfect square trinomials are always factorizable into \( (a + b)^2 \) or \( (a - b)^2 \), depending on the sign of the middle term. Recognizing and factoring these trinomials are key skills in algebra.

Binomial Squares

Binomial squares are expressions obtained from squaring a binomial, which is an algebraic expression containing two terms, such as \( (x + y) \) or \( (a - b) \). When we square a binomial, we use the formula \( (a \pm b)^2 = a^2 \pm 2ab + b^2 \), where the \( \pm \) symbol indicates that we maintain the sign of the middle term from the original binomial.

For example, squaring the binomial \( (x + 5) \) yields the binomial square \( x^2 + 10x + 25 \). This process involves applying the distributive property, also known as FOIL (First, Outer, Inner, Last), to multiply the terms. The resulting expression is a perfect square trinomial because it follows the pattern mentioned above.

Understanding binomial squares is beneficial for factoring trinomials because it helps us quickly identify when a trinomial can be factored back into the square of a binomial. It also lays the foundation for more advanced algebraic concepts like completing the square, a technique used to solve quadratic equations.