Problem 18 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ x^{2}+4 x-21 $$

Step-by-Step Solution

Answer

Answer: $$(x-3)(x+7)$$

Step 1: Identify the coefficients of the constant and middle term

In the given trinomial, $$x^{2}+4 x-21$$, the constant coefficient is -21 and the middle term coefficient is 4.

Step 2: Find the factors of the constant term

Next, we will find the factors of -21. Since -21 is a negative number, we will need to find two numbers, one positive and the other negative, whose product is -21. The possible factors of -21 are: (-1, 21), (1, -21), (-3, 7), and (3, -7).

Step 3: Find the pair of factors whose sum matches the middle term coefficient

Now we need to find the pair of factors from the list above whose sum matches our middle term coefficient (4). Going through the list, we can see that the pair (-3, 7) provides the desired sum: (-3) + 7 = 4.

Step 4: Write the trinomial as a product of binomials

Using the pair of factors found in Step 3, we can write our trinomial as a product of two binomials: $$(x-3)(x+7)$$ So the factored form of the trinomial $$x^{2}+4 x-21$$ is $$(x-3)(x+7)$$.

Key Concepts

Polynomial Factorization

Polynomial factorization is an essential concept in algebra that involves breaking down a polynomial into a product of simpler polynomials known as factors. For instance, when we have a quadratic trinomial like
\( x^2 + 4x - 21 \),
the aim is to find two binomials that, when multiplied together, return to the original trinomial. The process starts with identifying the constant and middle terms, and finding their possible factor pairs. One must consider the signs and find the pair whose sum or difference, depending on the case, will give the middle term's coefficient. Following that, we re-write the trinomial as the product of these binomials.
It's insightful to recognize that factoring is akin to reversing the process of expanding binomials where you use the distributive property (also known as the FOIL method) to expand products like \((x + a)(x + b)\) into a trinomial form \(x^2 + (a+b)x + ab\). Factoring can at times require trial and error, particularly when the coefficients are large or when the polynomial is not factorable using integers, which leads to being 'prime'.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. In the case of the trinomial \(x^2 + 4x - 21\), we have a quadratic expression, a specific type of algebraic expression where the highest power of the variable is 2. Algebraic expressions don't have an equals sign, unlike equations.
To manipulate these expressions, one needs to be familiar with operations such as addition, subtraction, multiplication, and sometimes division. For instance, when we factor \(x^2 + 4x - 21\), we're essentially looking for an expression that simplifies the original. We seek factors that will not change the value of the expression, only its form. This pivotal aspect of algebra aids in solving equations and can simplify complex problems. A strong conceptual grasp of algebraic expressions and their manipulation is indispensable for students to succeed in algebra and beyond.

Binomial Products

The product of two binomials forms the cornerstone of understanding polynomial multiplication. Binomials are algebraic expressions containing two terms, such as \((x - 3)\) and \((x + 7)\). When we multiply these, we get a trinomial.
To multiply two binomials, you can use the distributive property, commonly referred to as FOIL (First, Outer, Inner, Last). For example:\[ (x - 3)(x + 7) = x^2 - 3x + 7x - 21 \]Here, you multipliy the first terms of each binomial, then the outer terms, the inner terms, and finally, the last terms. The middle terms (-3x and 7x) combine to give 4x, resulting in the trinomial \(x^2 + 4x - 21\). This process works in reverse when factoring a trinomial, and understanding binomial products aids immensely in recognizing the patterns that arise during factorization. When factoring, it's important to check your work by multiplying the binomials to ensure they yield the original trinomial.