Problem 2 - Precalculus Mathematics for Calculus

Question

To factor the trinomial \(x^{2}+7 x+10\), we look for two integers whose product is _____and whose sum is _____ These integers are_____ and so the trinomial factors as_____

Step-by-Step Solution

Answer

The integers are 2 and 5; the trinomial factors as \((x + 2)(x + 5)\).

Step 1: Identify the Coefficients

In the trinomial \(x^2 + 7x + 10\), the coefficients are straightforward to identify. The coefficient of \(x^2\) is 1, the coefficient of \(x\) is 7, and the constant term is 10.

Step 2: Determine the Product and Sum

We need two integers whose product is equal to the coefficient of \(x^2\) (which is 1) times the constant term (10), giving us 10. Therefore, we seek two integers whose product is 10 and whose sum is the coefficient of \(x\), which is 7.

Step 3: List Pairs of Factors of 10

List the pairs of integers that multiply to 10: \((1, 10)\), \((2, 5)\), \((5, 2)\), and \((10, 1)\). Notice that we must also consider the signs, but since all coefficients are positive, we only consider positive numbers.

Step 4: Choose the Correct Pair

From our list, the pair that has a product of 10 and a sum of 7 is \((2, 5)\). Thus, 2 and 5 are the integers we seek.

Step 5: Write the Factored Form

Using the integers from the previous step, we can express the trinomial \(x^2 + 7x + 10\) in factored form as \((x + 2)(x + 5)\). This is because expanding \((x + 2)(x + 5)\) results in the original trinomial: \(x^2 + 5x + 2x + 10\) simplifies to \(x^2 + 7x + 10\).

Key Concepts

Quadratic Expressions

A quadratic expression is a type of polynomial that is degree two, which always contains a term with a variable raised to the second power. In simpler terms, it's an equation of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. For example, \(x^2 + 7x + 10\) is a quadratic expression. To understand them better:

• The term with \(x^2\) is the quadratic term. It defines the parabola's direction (upward or downward) when graphed.
• The term with \(x\) is the linear term, which influences the slope of the parabola.
• The constant term \(c\) moves the graph up or down on the y-axis.

Recognizing these terms helps in identifying how each part affects the expression's graph and assists in factorization.

Trinomial Factorization

Trinomial factorization involves rewriting a quadratic expression as a product of two binomials. This can make solving equations simpler, as well as providing insight into the roots of the equation.For the trinomial \(x^2 + 7x + 10\), we are interested in finding two numbers that multiply to the last term (10) and add up to the middle coefficient (7). These numbers here are 2 and 5.This factorization process means:

• The expression \(x^2 + 7x + 10\) becomes \((x + 2)(x + 5)\).
• The factors \((x + 2)\) and \((x + 5)\) when multiplied give back the original trinomial.
• Each binomial factor corresponds to a root of the equation \(x^2 + 7x + 10 = 0\).

Factoring trinomials is essential in algebra to simplify expressions, solve quadratic equations, and understand the relationships between components of the expression.

Algebraic Manipulation

Algebraic manipulation is the process of using mathematical operations to rearrange or simplify expressions and equations. It allows us to solve problems like factoring trinomials by applying systematic methods.While factoring \(x^2 + 7x + 10\), we follow these steps:

• Identify the coefficients to determine which operations are needed. In our example, look at the coefficients of 1, 7, and 10.
• Find two numbers whose product equals 10 (the constant term) and whose sum equals 7 (the linear coefficient).
• Rearrange the expression in a way that makes it easier to factor, using patterns and mathematical truths.

Effective algebraic manipulation is key in solving many algebra problems, working consistently with factoring strategies, and gaining deeper insight into how equations behave.