Problem 2 Factor completely. Identify any ... [FREE SOLUTION] (2024)

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Chapter 6: Problem 2

Factor completely. Identify any prime polynomials. $$ 2 x^{2}-18 $$

Short Answer

Expert verified

2(x+3)(x-3)

Step by step solution

02

Factor out the GCF

Next, factor out the GCF from each term in the polynomial. This means we divide each term by 2: 2(x^{2} - 9).

03

Factor the Quadratic Expression

Now, factor the quadratic expression inside the parenthesis, x^{2} - 9. Notice that this is a difference of squares which can be factored as: (x+3)(x-3).

04

Combine All Factors

Combine the GCF with the factored quadratic expression to get the completely factored form: 2(x+3)(x-3).

Key Concepts

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

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that can evenly divide all terms of a polynomial.
In our exercise, we have the polynomial \(2x^2 - 18\). We need to identify the GCF of the coefficients 2 and -18.
Since both numbers share the factor 2, it is the GCF.
Factoring out the GCF simplifies the polynomial by removing this factor from each term. For our polynomial, it turns into \(2(x^2 - 9)\).
This step is crucial because it makes the next steps of factoring much easier.

Difference of Squares

The difference of squares is a specific type of quadratic expression. It takes the form \(a^2 - b^2\).
This can always be factored into \((a+b)(a-b)\).
In our exercise, we reach the expression \(x^2 - 9\) after factoring out the GCF. Notice that this can be rewritten as \(x^2 - 3^2\), fitting the difference of squares format.
Applying the difference of squares rule, \(x^2 - 9\) becomes \((x+3)(x-3)\). This step exploits the structure of certain quadratics and offers a straightforward factoring method.

Quadratic Expressions

Quadratic expressions are polynomials that follow the form \(ax^2 + bx + c\). In many cases, they can be factored into products of linear binomials.
In our problem, once we factor out the GCF, we are left with \(x^2 - 9\), a simple quadratic with no linear term. This is a special case since it's also a difference of squares.
Generally, the approach involves: identifying patterns (like the difference of squares), factoring out the GCF first, and then breaking down the quadratic expression further.
For more complex quadratics, formulas such as the quadratic formula or completing the square might be used.
But in our exercise, recognizing the difference of squares allowed a simpler solution.

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Problem 2 Factor completely. Identify any ... [FREE SOLUTION] (3)

Most popular questions from this chapter

Simplify \(\frac{108}{124}\) into lowest terms.Factor completely. Identify any prime polynomials. $$ 25 w^{3}-10 w^{2}+w $$A student said that the solutions of \((x+2)(x-6)=15\) are \(x=-2\) and \(x=6\),since \(-2+2=0\) and \(6-6=0\). Explain what is wrong with this thinking.Factor completely. Identify any prime polynomials. $$ 3 p-3 z^{2} $$
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Problem 2 Factor completely. Identify any ... [FREE SOLUTION] (2024)

FAQs

What if I don't have my phone for 2 step verification? ›

If you've lost access to your primary phone, you can verify it's you with: Another phone signed in to your Google Account. Another phone number you've added in the 2-Step Verification section of your Google Account. A backup code you previously saved.

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The Microsoft Authenticator app requires an internet connection to send verification codes. Check if the app has the necessary permissions to send notifications. On your mobile device, go to Settings > Apps > Microsoft Authenticator > Permissions, and ensure that the app has permission to send notifications.

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If you've lost access to your 2FA device, you can recover your account by using backup codes, alternative recovery options like a secondary email or phone number, or by contacting customer support. Be ready to confirm your identity by answering a few security questions or providing proof of ID.

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Turn off 2-Step Verification
  1. Open your device's Settings app and tap Google. Manage your Google Account.
  2. At the top, tap Security.
  3. Under "How you sign in to Google," tap 2-Step Verification. You might need to sign in.
  4. Tap Turn off.
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What if I can't access my two-factor authentication code? ›

If you have forgotten your password and you've lost access to your two-factor authentication credentials, you can start account recovery to regain access to your account. You'll need to verify your identity using a recovery authentication factor, such as an SSH key or previously verified device.

How do I fix not receiving verification code? ›

How to Fix It When You're Not Receiving Verification Code Texts on Android
  1. Reboot your Android. ...
  2. Check your phone number. ...
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  8. Update Android.
Jul 26, 2023

How to get SMS verification code without phone? ›

Another option is to use virtual phone numbers provided by various online services. These virtual numbers can be used to receive SMS codes without using a physical phone. You can access these virtual phone numbers through apps or websites and receive SMS codes for verification purposes.

How do I fix Microsoft Authenticator problem? ›

If you're using mobile data, try switching to Wi-Fi and vice-versa. Make sure Airplane mode is off. Make sure you're using the latest version of Authenticator - Microsoft does not support any app versions more than 12 months old. Tap Settings and make sure App updates is turned on.

How to retrieve 2 factor authentication? ›

Recover an account
  1. Sign in to your Google Admin console. ...
  2. In the Admin console, go to Menu Directory. ...
  3. Click the user you want in the list. ...
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  5. Click 2-step verification. ...
  6. Click Get Backup Verification Codes.
  7. Copy one of the verification codes.
  8. Send the backup code to the user in an IM or text message.

What are the solution for broken authentication? ›

Solutions include implementing Multi-Factor Authentication (MFA), enforcing strong password policies, limiting failed login attempts, securing session management, and regular security audits.

How do I get two-factor authentication without a phone? ›

You don't need to have a smartphone, as you can still use Google 2FA with:
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How do I turn off two-factor authentication? ›

Turn off 2-Step Verification
  1. Open your Google Account.
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  3. Select Turn off.
  4. A pop-up window will appear to confirm that you want to turn off 2-Step Verification. Select Turn off.

How do you solve factor problems? ›

The Solve by Factoring process will require four major steps:
  1. Move all terms to one side of the equation, usually the left, using addition or subtraction.
  2. Factor the equation completely.
  3. Set each factor equal to zero, and solve.
  4. List each solution from Step 3 as a solution to the original equation.

How do you factor completely? ›

To factor completely:
  1. Search for a greatest common factor. ...
  2. If the leading coefficient is negative, factor out the negative.
  3. Examine what remains, looking for a trinomial or a binomial which can be factored. ...
  4. Express the answer as the product of all of the factors you have found.
  5. Check to see if your factors are correct.

How to get 2-Step Verification code without phone? ›

You don't need to have a smartphone, as you can still use Google 2FA with:
  1. Passcodes delivered to your phone by text message or automated phone call. ...
  2. Google prompts sent to an iPad or Android tablet.
  3. An authenticator app running on an iPad or Android tablet.
  4. A security key.
  5. Backup codes.

How do I get through two-step authentication if I changed my phone number? ›

To change the phone number for 2-Step Verification on the mobile apps:
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  3. Select "Password & Security"
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Can you do two-factor authentication without a phone? ›

Using 2FA without a mobile device

Assuming that a computer is used, as two-factor authentication without a mobile device and computer would make little sense, the following two options are available: Installing an authenticator app directly on the desktop computer or notebook. Using a security key.

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  1. Alternative Methods for SMS Verification without Phone Numbers. ...
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Apr 16, 2024

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