Factor the expression: 2a 11z +5

Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Factoring Quadratic Expressions

In this lesson, we will learn how to factor quadratic expressions. Quadratic expressions take the form \(ax^2 + bx + c\). The process of factoring involves breaking down the expression into products of simpler expressions.

#### Example Problem

**Factor the expression:**

\[ 2x^2 - 11x + 5 \]

To factor this quadratic expression, follow the steps below:

1. **Find two numbers that multiply to \(ac\)** (the product of the leading coefficient and the constant term) and add to \(b\) (the middle coefficient).
    - Here, \(a = 2\), \(b = -11\), and \(c = 5\).
    - Thus, we need two numbers that multiply to \(2 \cdot 5 = 10\) and add to \(-11\).

2. **Numbers that fit are \(-1\) and \(-10\)**. Therefore, we rewrite the middle term using these numbers.
    \[ 2x^2 - x - 10x + 5 \]

3. **Group the terms to factor by grouping**.
    - Group the first two terms and the last two terms:
        \[ 2x^2 - x - 10x + 5 = (2x^2 - x) + (-10x + 5) \]

4. **Factor out the greatest common factor (GCF) from each group**.
    - From \((2x^2 - x)\), factor out \(x\):
        \[ x(2x - 1) \]
    - From \((-10x + 5)\), factor out \(-5\):
        \[ -5(2x - 1) \]

5. **Combine the factored groups**:
    - The expression becomes:
        \[ x(2x - 1) - 5(2x - 1) \]

6. **Factor out the common binomial**:
    \[ (2x - 1)(x - 5) \]

So, the factored form of \( 2x^2 - 11x + 5 \) is \( (2x - 1)(x - 5) \).

#### Summary

To factor a quadratic expression \(ax^2 + bx + c\):
1. Identify
Transcribed Image Text:### Factoring Quadratic Expressions In this lesson, we will learn how to factor quadratic expressions. Quadratic expressions take the form \(ax^2 + bx + c\). The process of factoring involves breaking down the expression into products of simpler expressions. #### Example Problem **Factor the expression:** \[ 2x^2 - 11x + 5 \] To factor this quadratic expression, follow the steps below: 1. **Find two numbers that multiply to \(ac\)** (the product of the leading coefficient and the constant term) and add to \(b\) (the middle coefficient). - Here, \(a = 2\), \(b = -11\), and \(c = 5\). - Thus, we need two numbers that multiply to \(2 \cdot 5 = 10\) and add to \(-11\). 2. **Numbers that fit are \(-1\) and \(-10\)**. Therefore, we rewrite the middle term using these numbers. \[ 2x^2 - x - 10x + 5 \] 3. **Group the terms to factor by grouping**. - Group the first two terms and the last two terms: \[ 2x^2 - x - 10x + 5 = (2x^2 - x) + (-10x + 5) \] 4. **Factor out the greatest common factor (GCF) from each group**. - From \((2x^2 - x)\), factor out \(x\): \[ x(2x - 1) \] - From \((-10x + 5)\), factor out \(-5\): \[ -5(2x - 1) \] 5. **Combine the factored groups**: - The expression becomes: \[ x(2x - 1) - 5(2x - 1) \] 6. **Factor out the common binomial**: \[ (2x - 1)(x - 5) \] So, the factored form of \( 2x^2 - 11x + 5 \) is \( (2x - 1)(x - 5) \). #### Summary To factor a quadratic expression \(ax^2 + bx + c\): 1. Identify
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