Question
upstudy study bank question image url

Factor the trinomial \[ 2 x^{2} z+20 x z+42 z \] Seleci the correct choice below and, if necessary, fil in the answer box to complete your choice. A. \( 2 x^{2} z+20 x z+42 z= \) (Factor completey) B. \( 2 x^{2} z+20 x z+42 z \) is prime Helo mo solun this

Ask by Carroll Chambers. in the United States
Mar 11,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The trinomial factors to \(2z(x + 3)(x + 7)\).

Solution

We start with the expression \[ 2x^2z + 20xz + 42z. \] **Step 1: Factor out the greatest common factor (GCF).** Notice that each term includes the factor \(z\) and is even, so we can factor out \(2z\): \[ 2x^2z + 20xz + 42z = 2z\left(x^2 + 10x + 21\right). \] **Step 2: Factor the quadratic \(x^2 + 10x + 21\).** We look for two numbers that multiply to \(21\) and add to \(10\). The numbers \(3\) and \(7\) work because \[ 3 \times 7 = 21 \quad \text{and} \quad 3 + 7 = 10. \] Thus, the quadratic factors as \[ x^2 + 10x + 21 = (x+3)(x+7). \] **Final Answer:** \[ 2x^2z + 20xz + 42z = 2z(x+3)(x+7). \] Therefore, the correct choice is: A. \( 2x^2z + 20xz + 42z = 2z(x+3)(x+7) \) (Factor completely)

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Mind Expander

To factor the trinomial \( 2x^{2}z + 20xz + 42z \), we can start by looking for a common factor in all the terms. Notice that \( z \) can be factored out: \[ 2x^{2}z + 20xz + 42z = z(2x^{2} + 20x + 42) \] Next, we can focus on factoring the quadratic \( 2x^{2} + 20x + 42 \). First, we can find the greatest common factor (GCF) of the coefficients 2, 20, and 42, which is 2: \[ 2(x^{2} + 10x + 21) \] Now we need to factor the quadratic expression \( x^{2} + 10x + 21 \). We can look for two numbers that multiply to 21 and add up to 10. Those numbers are 3 and 7: \[ x^{2} + 10x + 21 = (x + 3)(x + 7) \] Putting it all together, we have: \[ 2(x + 3)(x + 7) \] So, the complete factorization of the original trinomial is: \[ 2x^{2}z + 20xz + 42z = z \cdot 2(x + 3)(x + 7) \] Thus, the answer for part A is: \( z \cdot 2(x + 3)(x + 7) \)

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy