Problem 74 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(u^{2}+101 u+100\)

Step-by-Step Solution

Answer

(u + 1)(u + 100)

Step 1: Identify coefficients

Identify the coefficients in the trinomial. Here, the coefficient of the linear term (b) is 101, and the constant term (c) is 100.

Step 2: Find factor pairs

Find pairs of factors of the constant term (100) that add up to the linear coefficient (101). For 100, the factor pairs are (1, 100) because 1 + 100 = 101.

Step 3: Write the factorization

Using the pair found in Step 2, write the factorization of the trinomial. The trinomial can be factored as (u + 1)(u + 100).

Key Concepts

Polynomial Factorization

Polynomial factorization is a way to express a polynomial as the product of simpler polynomials. This concept helps in simplifying expressions and solving polynomial equations.
In our example, the polynomial given is a trinomial, which means it has three terms. It is essential to recognize and factorize these polynomials, as it helps in understanding roots and solving equations.

To factor a polynomial effectively, follow these steps:

• Identify the polynomial's structure.
• Find common factors for the terms.
• Use factoring techniques such as grouping or special factorizations like the difference of squares.

In our exercise, we factor the trinomial by looking for factors of the constant term that add up to the coefficient of the linear term.

Trinomial Factoring

Trinomial factoring refers to the specific process of factorizing a polynomial with three terms. The standard form is usually given as:
\[ax^{2} + bx + c\]
In our case, the trinomial is: \[u^{2} + 101u + 100\]. To factor it, follow these steps:

• Identify the coefficients: Here, a=1, b=101, and c=100.
• Find two numbers that multiply to c (100) and add to b (101).
• For 100, the factor pairs are (1, 100) because 1 + 100 = 101; these pairs are used to break down and simplify the trinomial.

Finally, rewrite the trinomial as the product of two binomials: \[(u + 1)(u + 100)\]. This process can be simplified by practice and understanding the relationships between coefficients.

Quadratic Equations

Quadratic equations are equations of the form \[ax^{2} + bx + c = 0\]. These equations can have real or complex solutions based on the discriminant \[b^{2} - 4ac\].
Solving quadratic equations can be achieved using different methods, including:

• Factoring: As in our example, if a quadratic can be factored into binomials, its solutions can be found by setting each binomial to zero and solving.
• Quadratic Formula: \[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\] is useful when factoring is difficult.
• Completing the Square: This method modifies the equation to make it easier to solve by creating a perfect square trinomial.

Understanding how to factor and solve these equations helps in many areas of algebra and is fundamental to further studies in mathematics.