Problem 44 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ x^{2}+4 x-10 $$

Step-by-Step Solution

Answer

The trinomial has no simple factorization involving real numbers.

Step 1: Identify the trinomial structure

The given trinomial is \(x^2 + 4x - 10\). It is in the standard quadratic form \(ax^2 + bx + c\), where \(a=1\), \(b=4\), and \(c=-10\).

Step 2: Check for a Greatest Common Factor (GCF)

Examine the coefficients and terms of the trinomial \(x^2 + 4x - 10\) to see if there is a GCF other than 1. In this case, the coefficients 1, 4, and -10 have no common factor other than 1, so the GCF is 1.

Step 3: Attempt to factor the trinomial

Since the GCF is 1, proceed to factor the quadratic trinomial itself. Look for two numbers whose product is \(a \cdot c = 1 \cdot (-10) = -10\) and whose sum is \(b = 4\).

Step 4: Identify factor pairs

The factor pairs of \(-10\) are \((-1, 10), (1, -10), (-2, 5), (2, -5)\). Check which pair adds to \(4\).

Step 5: Choose the correct factor pair

The pair \((-2, 5)\) works because \(-2 + 5 = 3\). However, there is a typographical error as it should yield the sum \(4\). Review your steps or calculations.

Step 6: Consider alternative methods

Since traditional factoring isn't working, a typo might have emerged, causing misidentification. Examining each case confirms no factorization would yield sufficient results without certain numerical predictions. Hence, quadratic methods might be validated otherwise—tho finding a solution involves different mathematical aspects beyond factorization.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2, which means it includes a term that is squared. It has the general form: \[ ax^2 + bx + c = 0 \]where:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

Quadratic equations can have up to two solutions because the highest power of the variable is 2. These solutions are often referred to as the roots of the equation. In the case of factoring trinomials, you're essentially looking for these solutions, expressed in factored form rather than through a series of computations. Factoring allows you to express the quadratic equation in terms of two binomials.

Greatest Common Factor

The Greatest Common Factor (GCF) is an important concept when factoring trinomials because it’s always good practice to factor it out first if it's greater than 1. The GCF is the largest positive integer that divides each of the coefficients in a polynomial. By removing the GCF, we make the equation simpler and easier to work with.
In the given trinomial \[ x^2 + 4x - 10 \]we checked the coefficients which were 1, 4, and -10, and found that they do not share any common factor other than 1. This means the GCF here is 1, and thus does not need to be extracted. Always remember: Checking for a GCF is the first step before any further factoring attempts.

Factor Pairs

Factor pairs are pairs of numbers that, when multiplied together, give a particular product. Finding the right factor pairs is central to successfully factoring a quadratic trinomial.
To factor \( x^2 + 4x - 10 \) we multiply the leading coefficient \( a = 1 \) by the constant term \( c = -10 \), resulting in -10. We need to find two numbers that multiply to -10, yet add up to the middle term coefficient, which is 4. The factor pairs of -10 are:

• (-1, 10)
• (1, -10)
• (-2, 5)
• (2, -5)

Examining these pairs, none satisfies both conditions to sum to 4 whilst having their product as -10. For this exercise, traditional factor pairs do not yield a solution; hence, alternative methods like the quadratic formula need to be applied.

Quadratic Formula

When traditional factoring methods fail, the quadratic formula is a reliable alternative. It gives the roots of any quadratic equation in the form \[ ax^2 + bx + c = 0 \]The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula will solve a quadratic equation by calculating both possible values of \( x \) using the given coefficients.
In scenarios like \( x^2 + 4x - 10 \), if factor pairs can't be found, this formula ensures a path to the roots. The key is focusing on the "discriminant" \( b^2 - 4ac \)under the square root which tells us about the nature of the roots:

• If positive, there are two distinct real roots.
• If zero, there are exactly one real root.
• If negative, the roots are complex.

Understanding the quadratic formula is crucial, since its use extends beyond simple factorization and solves all quadratic problems.