Problem 103 Factor each polynomial completel... [FREE SOLUTION]

Chapter 5: Problem 103

Factor each polynomial completely. $$ -2 w^{2}+18 w+20 $$

Short Answer

Expert verified

$ -2(w - 10)(w + 1) $

Step by step solution

- Identify the Greatest Common Factor

Look for the greatest common factor (GCF) of the given polynomial terms $ -2w^2, 18w, \text{and} 20 $. The GCF for these terms is $ 2 $.

- Factor Out the GCF

Factor out $ -2 $ from each term: $ -2(w^2 - 9w - 10) $.

- Factor the Remaining Quadratic Expression

Rewrite the quadratic expression inside the parentheses: $ -2(w^2 - 9w - 10) $. We need to find two numbers that multiply to $-10$ and add up to $-9$. These numbers are $ -10 $ and $ 1 $.

- Write the Factored Form

Use the numbers found in Step 3 to write the factored form of the quadratic: $ -2(w - 10)(w + 1) $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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Identifying the greatest common factor (GCF) is a crucial step in factoring polynomials. The GCF is the largest factor that divides all the terms in a polynomial. For the given polynomial $ -2w^2 + 18w + 20 $, the terms are $ -2w^2 $, $ 18w $, and $ 20 $. To determine the GCF, follow these steps:

List the factors of each term.
- For $ -2w^2 $: the factors are $ -2, w, w $.
- For $ 18w $: the factors are $ 2, 3, 3, w $.
- For $ 20 $: the factors are $ 2, 2, 5 $.
Identify the common factors across all terms: In this case, it鈥檚 just $ 2 $.
The GCF is $ 2 $.

By factoring out the GCF, you simplify the polynomial, making it easier to work with in later steps. In this exercise, after identifying the GCF, we factored it out:
$ -2w^2 + 18w + 20 = -2( w^2 - 9w - 10 ) $

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A quadratic expression is a polynomial of degree 2, generally written in the form $ ax^2 + bx + c $. For this specific problem:

We have the quadratic expression inside the parentheses: $ w^2 - 9w - 10 $.
The standard form here is with $ a = 1 $, $ b = -9 $, and $ c = -10 $.
To factor it, we need to find two numbers that:
- Multiply together to $ ac $, which is $ 1 * -10 = -10 $
- Add up to $ b $, which is $ -9 $.
Those two numbers are $ -10 $ and $ 1 $, because:
- $ -10 * 1 = -10 $
- $ -10 + 1 = -9 $

Thus, the quadratic expression $ w^2 - 9w - 10 $ can be factored into $ (w - 10)(w + 1) $.

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To express a polynomial in its factored form means to write it as a product of simpler expressions. After identifying and factoring out the greatest common factor and handling the quadratic expression, the final steps are straightforward:

From our given polynomial $ -2w^2 + 18w + 20 $, we factored out $ -2 $. Thus, we have: $ -2(w^2 - 9w - 10) $.
Next, we factored the quadratic expression $ w^2 - 9w - 10 $ into $ (w - 10)(w + 1) $.
Putting it all together, the complete factored form of the polynomial is: $ -2(w - 10)(w + 1) $.

Understanding these steps is crucial for moving forward in algebra, as factoring polynomials is a foundational skill in solving various types of algebraic equations. Once you have the polynomial in factored form, you can use it for solving equations, graphing parabolas, and more.

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91影视

Factor each polynomial completely. $$ -2 w^{2}+18 w+20 $$

Short Answer

Step by step solution

- Identify the Greatest Common Factor

- Factor Out the GCF

- Factor the Remaining Quadratic Expression

- Write the Factored Form

Key Concepts

headline of the respective core concept

headline of the respective core concept

headline of the respective core concept

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