Problem 17 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ a^{2}+3 a-4 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial is $$(a-1)(a+4)$$.

Step 1: Identify the factors of the constant term

Find the factors of the constant term, -4. The pairs of factors are (-1, 4) and (1, -4).

Step 2: Determine the correct factors to sum up to the linear term coefficient

Now, we want to find the pair of factors of -4 that sum up to the coefficient of the linear term, which is 3. In this case, the correct pair is (-1, 4) because -1 + 4 = 3.

Step 3: Write the factored form

We found that the pair (-1, 4) satisfies both conditions. So the factored form will be: (a-1) (a+4)

Step 4: Final Answer

The factored form of the trinomial is $$ (a-1)(a+4) $$

Key Concepts

Polynomial Factorization

Factoring trinomials is an essential part of polynomial factorization, which helps to simplify and solve polynomial equations. When we factor a polynomial, we are essentially expressing it as a product of simpler polynomials. This process is somewhat similar to factoring numbers, where we decompose a number into its prime factors. In the context of the polynomial \( a^2 + 3a - 4 \), we'd like to express it in the form of \((a + m)(a + n)\). By understanding the relationships between the coefficients and the constant term, we can break down complex expressions into simpler, more manageable ones.

Here's a more intuitive way to think about it: each term in the trinomial is related. The constant term, in our example \(-4\), is the product of the numbers \(m\) and \(n\), and the linear term coefficient \(3\) is the sum of \(m\) and \(n\). By working through this step-by-step, we can find appropriate values for \(m\) and \(n\) that satisfy both conditions.

Quadratic Equations

Quadratic equations are a particular type of polynomial equation characterized by the term \(ax^2\) where \(a eq 0\). They follow the standard form \(ax^2 + bx + c\). In our exercise, the quadratic equation \(a^2 + 3a - 4\) is factored as \((a-1)(a+4)\). This form allows us to solve the equation by setting each factor equal to zero.

Solving quadratic equations through factoring involves these steps:

• Express the quadratic in standard form.
• Factor the quadratic expression into two binomials.
• Set each binomial equal to zero to solve for the variable.

For example, from the factored form \((a-1)(a+4)=0\), we solve for \(a\) by setting \(a-1=0\) or \(a+4=0\), resulting in solutions \(a=1\) and \(a=-4\). These solutions represent the points where the quadratic graph intersects the x-axis.

Algebraic Expressions

Algebraic expressions, like \(a^2 + 3a - 4\) in our exercise, are combinations of variables, coefficients, and constants linked together by operations such as addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions is key to solving a wide range of algebra problems.

Here are some key insights about dealing with algebraic expressions:

• A trinomial is an algebraic expression consisting of three terms.
• Each term is separated by either an addition or subtraction sign.
• Factoring transforms a complex expression into a simpler product of factors.

By mastering techniques such as factoring, students gain a powerful tool for simplifying expressions, solving equations, and even verifying their work in other mathematical contexts. It helps in identifying the roots of equations, making algebraic computations more efficient and manageable.