Problem 25 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ a^{2}+a-30 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial $$ a^2 + a - 30 $$ is $$ (a + 5)(a - 6) $$.

Step 1: Identify Quadratic Trinomial

The given trinomial is a quadratic trinomial, of the form $$ ax^2 + bx + c $$. In this case, we have $$ a = 1 $$, $$ b = 1 $$, and $$ c = -30 $$.

Step 2: Find integers m, n that satisfy the conditions

For a quadratic trinomial, we want to find two integers, m and n, such that $$ m \times n = ac $$ and $$ m + n = b $$. In our case, $$ ac = 1 \times (-30) = -30 $$ and $$ b = 1 $$. Now, let's find possible factors of -30: 1. 1 and -30 2. -1 and 30 3. 2 and -15 4. -2 and 15 5. 3 and -10 6. -3 and 10 7. 5 and -6 8. -5 and 6 Out of these pairs, we see that the pair (5, -6) satisfies the condition: $$ m + n = 5 + (-6) = -1 $$.

Step 3: Factor the trinomial

Now that we've found the required integers, m = 5 and n = -6, we can factor the trinomial using these integers. Our factored trinomial will be in the form: $$ (a + m)(a + n) $$ So, factoring the given trinomial, we get: $$(a + 5)(a - 6)$$ Thus, the factored form of the given trinomial $$ a^2 + a - 30 $$ is $$ (a + 5)(a - 6) $$.

Key Concepts

Quadratic Trinomials

A quadratic trinomial is an algebraic expression that contains three terms and the highest exponent of the variable is 2. In general, the format of a quadratic trinomial is given by \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. Understanding quadratic trinomials is important because they are commonly used in equations that must be factored or solved for their roots.

When we talk about quadratic trinomials, it's essential to highlight their components. The term \( ax^2 \) is known as the quadratic term, \( bx \) is the linear term, and \( c \) is the constant term. In the example \( a^2 + a - 30 \), we see it follows the trinomial format with \( a = 1 \), \( b = 1 \), and \( c = -30 \). Recognizing this format helps us understand how to approach factoring it into simpler components.

Factored Form

Factored form is the representation of a polynomial as a product of its factors. For quadratic trinomials, factoring involves rewriting the trinomial as the product of two binomials. This is an important concept as it simplifies expressions and aids in solving quadratic equations.

The process to achieve the factored form typically involves finding two numbers whose product is equal to \( a \times c \) (where \( a \) and \( c \) are coefficients from \( ax^2 + bx + c \)) and whose sum is equal to \( b \). In our example, \( a^2 + a - 30 \) factors into \((a + 5)(a - 6)\).

The result signifies that the product of \((a+5)\) and \((a-6)\) will give us back the original trinomial \(a^2 + a - 30\). Seeing the polynomial in its factored form tells us the values of \( a \) that make the polynomial equal zero (the solutions), also known as the roots.

Integer Factorization

Integer factorization in the context of factoring trinomials involves finding two integers that multiply to the product of the leading coefficient \( a \) and the constant term \( c \), and add up to the middle coefficient \( b \). This technique is crucial in simplifying the trinomial into a product of two binomials.

In our specific problem, we're tasked with factoring \( a^2 + a - 30 \). To do this, we find integers \( m \) and \( n \) such that \( m \times n = -30 \) and \( m + n = 1 \). Using integer factorization, we examine factor pairs of \(-30\) and determine that the pair \( (5, -6) \) meets the criteria: \( 5 + (-6) = -1 \).

• Begin by listing pairs: \( (1, -30), (-1, 30), (2, -15), (-2, 15), (3, -10), (-3, 10), (5, -6), (-5, 6) \).
• Select the pair that adds to \( b \), which is \( 1 \) in this case.

This straightforward approach not only helps in factoring the trinomial but also plays a key role when solving or simplifying quadratic equations.