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Magazine Chapter 0: Problem 34
Factor the expression completely. \(x^{2}+3 x-10\)
Short Answer
Expert verified
The expression is factored as \((x + 5)(x - 2)\).
Step by step solution
01
Identify the Coefficients
For the quadratic expression \(x^2 + 3x - 10\), identify the coefficients: \(a = 1\), \(b = 3\), and \(c = -10\). These correspond to the general form \(ax^2 + bx + c\).
02
Find Two Numbers that Multiply to c and Add to b
We need two numbers that multiply to \(-10\) (the constant \(c\)) and add up to \(3\) (the linear coefficient \(b\)). The numbers are \(5\) and \(-2\) because \(5 \times (-2) = -10\) and \(5 + (-2) = 3\).
03
Rewrite the Middle Term
Rewrite the middle term \(3x\) as the sum of two terms using the numbers found in Step 2: \(x^2 + 5x - 2x - 10\).
04
Group the Terms
Group the terms into two pairs: \((x^2 + 5x)\) and \((-2x - 10)\).
05
Factor by Grouping
Factor out the greatest common factor from each pair. From \(x^2 + 5x\), factor \(x\): \(x(x + 5)\). From \(-2x - 10\), factor \(-2\): \(-2(x + 5)\).
06
Extract the Common Binomial Factor
Notice that \((x + 5)\) is common in both terms. Factor \((x + 5)\) out: \((x + 5)(x - 2)\).
07
Verify the Factorization
Expand the factored expression to verify correctness: \((x + 5)(x - 2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10\). The original expression is regained, confirming the factorization is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Trinomials
Factoring trinomials is an essential skill in algebra that involves expressing a trinomial as the product of two binomials. Trinomials are polynomials with three terms, commonly structured as \(ax^2 + bx + c\). The goal of factoring is to simplify or rewrite expressions to solve equations or to make them easier to handle in calculations.
- Step 1: Identify the coefficients \(a\), \(b\), and \(c\) from the trinomial. These are the leading coefficient, the middle coefficient, and the constant term, respectively.
- Step 2: Look for two numbers that multiply to the constant term \(c\) and add to the middle coefficient \(b\). This is the key observation to factor correctly.
- Step 3: Rewrite the trinomial using these two numbers to substitute the middle term, which helps to reveal a pattern for grouping.
Algebraic Expressions
An algebraic expression is a mathematical phrase that involves numbers, variables, and operation symbols like addition and multiplication. In the context of factoring, focusing on algebraic expressions refers to the simplification or transformation of a polynomial into a product of simpler expressions.
- They can represent real-world scenarios and are used extensively to model equations and inequalities.
- The expression \(x^2 + 3x - 10\) is an example where we use variables and constants together in terms of operations.
- Breaking down these into partial expressions helps in realizing factorization by grouping terms that share a common factor.
Polynomial Equation Solving
Solving polynomial equations often involves transforming the equation into a factored form. In this way, the roots or solutions can be readily identified. This process directly uses the concept of factoring trinomials.
- When a polynomial is expressed as a product of simpler polynomials, setting each factor equal to zero allows us to solve for the variable.
- In our example, once we represent \(x^2 + 3x - 10\) as \((x + 5)(x - 2)\), setting each binomial to zero gives logical roots: \(x + 5 = 0\) or \(x - 2 = 0\).
- This means \(x = -5\) or \(x = 2\) are our solutions. Identifying these solutions are crucial in understanding the behavior of quadratic relationships in functions.
