Problem 91 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(x^{2}+x+5\)

Step-by-Step Solution

Answer

The trinomial \(x^2 + x + 5\) cannot be factorized using integers.

Step 1 - Identify coefficients

Identify the coefficients of the given trinomial. In this case, the trinomial is given as \[ x^2 + x + 5. \]The coefficients are: - For \( x^2 \), the coefficient is equal to 1.- For x, the coefficient is also 1.- The constant term is 5.

Step 2 - Check for factorizability

To factorize the trinomial of the form \[ x^2 + bx + c \], look for two numbers that multiply to get the constant term (c) and add up to the coefficient of the middle term (b).Here we need two numbers that multiply to 5 and add to 1.

Step 3 - Analyze possible pairs

Analyze pairs of numbers that multiply to 5: - Pair (1, 5) : Their sum is 6.- Pair (-1, -5) : Their sum is -6.- Pair (5, -1) : Their sum is 4.- Pair (-5, 1) : Their sum is -4.

Step 4 - Conclusion for factorization

Since none of the pairs of numbers multiply to 5 and add to 1, it is concluded that the trinomial \[ x^2 + x + 5 \]cannot be factorized using integers.

Key Concepts

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Trinomials are special polynomials that consist of exactly three terms. These terms often take the form \(ax^2 + bx + c \), where each part of the expression serves a unique role.

The first term, \(ax^2\), is called the quadratic term because it involves \(x\) squared. The second term, \(bx\), is the linear term, due to its involvement with only \(x\). Finally, the third term, \c\, is the constant term, as it does not contain the variable \(x\).

Understanding trinomials is crucial because they frequently appear in algebra and beyond. Tackling a trinomial starts with identifying these parts and their respective coefficients.

headline of the respective core concept

Coefficients are the numerical values placed in front of variables in algebraic expressions. In the trinomial \(x^2 + x + 5\), the coefficients are essential for understanding the structure of the equation.

For example:

• In the term \(x^2\), the coefficient is 1 (since it is \(1x^2\))
• In the term \(x\), the coefficient is also 1
• The number 5 is a constant term and stands alone without a variable

Identifying the coefficients \b\ and \c\ is the first step in many algebraic processes, such as factorization. They provide the necessary values to proceed with further algebraic manipulation.

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Factorization is the process of breaking down a polynomial into simpler terms called factors. These factors, when multiplied together, give the original polynomial.

For the trinomial \(x^2 + x + 5\), factorization involves finding two binomials that multiply to the original trinomial. This is generally expressed as \((x + m)(x + n)\), where \m\ and \ are numbers that satisfy the condition \(m \cdot n = c\) and \(m + n = b\).

Following this method, we analyzed pairs of numbers that multiply to 5 and examined their sums. However, we found no pairs that fit both criteria. Thus, \(x^2 + x + 5\) is not factorable using integers. This conclusion is essential because it tells us that some trinomials may require other methods for simplification or may be irreducible.