Problem 22 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ t-90+t^{2} $$

Step-by-Step Solution

Answer

The expression factors to \((t - 9)(t + 10)\).

Step 1: Recognize the expression's form

The given expression is \( t - 90 + t^2 \). Rearrange it to standard quadratic form, giving \( t^2 + t - 90 \). This is a quadratic expression of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 1 \), and \( c = -90 \).

Step 2: Identify factor pairs of the constant term

To factor the quadratic, we look for two numbers that multiply to \( c = -90 \) and add to \( b = 1 \). The factor pairs of \(-90\) are: \((-1, 90), (-2, 45), (-3, 30), (-5, 18), (-6, 15), (-9, 10)\).

Step 3: Select the correct factor pair

Identify which factor pair sums to \( 1 \). Testing each pair: \(-9 + 10 = 1\). Hence, \((-9, 10)\) is the correct pair.

Step 4: Write the factors

Using the pair \((-9, 10)\), factor the expression as \((t - 9)(t + 10)\). This means the expression \( t^2 + t - 90 \) can be written as \((t - 9)(t + 10)\).

Step 5: Verify the factoring

Expand \((t - 9)(t + 10)\) to ensure it equals the original expression: \(t \cdot t + t \cdot 10 - 9 \cdot t - 9 \cdot 10 = t^2 + 10t - 9t - 90 = t^2 + t - 90\). The factoring is verified as correct.

Key Concepts

Quadratic Expressions

Quadratic expressions are a fundamental concept in algebra, providing the basis for solving various mathematical problems. A quadratic expression is a polynomial of degree two. This means it is an algebraic expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratics are used to model real-world phenomena, such as projectile motion or optimization problems, and are characterized by their parabolic graphs when plotted on a coordinate system.
When working with quadratic expressions, the main goal is often to factor them. This makes it easier to solve for zeros or roots, which are the values of \( x \) that satisfy the equation when the quadratic is set to zero.
In our exercise, the quadratic expression \( t^2 + t - 90 \) needs to be factored to solve for its roots. To do this, understanding the standard form and characteristics of quadratics is essential. Recognize that the coefficients \( a=1 \), \( b=1 \), and \( c=-90 \) guide you through the factoring process.

Factoring Techniques

Factoring is a powerful technique used to simplify algebraic expressions and solve equations. Several techniques exist for factoring quadratic expressions, making it crucial to choose the right one based on the given problem.

• **Recognizing Patterns**: Some quadratics can be factored using patterns like difference of squares or perfect square trinomials.
  
• **Trial and Error**: Finding two numbers that multiply to the constant term \( c \) and add to the linear coefficient \( b \). This method is useful when \( a = 1 \), as in our example.
  
• **Decomposition**: Splitting the middle term and factoring by grouping, generally used when \( a eq 1 \).
  

In our specific problem, finding factor pairs of \(-90\) that add up to \(1\) is crucial. By testing the factor pairs \((-9, 10)\) until we reach the solution, we can rewrite the expression as \((t - 9)(t + 10)\). This decomposition allows us to further solve the equation or simplify it as needed.

Algebraic Expressions

Algebraic expressions form the foundational building blocks of algebra, including terms, coefficients, and variables. These expressions allow us to represent mathematical relationships and perform manipulations to uncover new insights.
In algebra, expressions vary from simple monomials to complex polynomials. Quadratic expressions, like the one in our exercise, bridge these forms with their degree of two and involve multiple terms.
Knowing how to manipulate algebraic expressions is vital. This skill includes expanding, simplifying, and, significantly, factoring expressions. Factoring is particularly important as it simplifies problems and is a key step in many solution processes, such as solving equations or identifying properties like roots and intercepts.
Through understanding the composition and characteristics of algebraic expressions, we can leverage factoring and other techniques to explore the intricate relationships within mathematical problems, ultimately leading to solutions and conclusions.