As a registered tutor on UrbanPro.com specializing in Class 7 Tuition, I understand the importance of providing clear and structured explanations for mathematical problems. Let's delve into the factorization of the given polynomial expression:

Problem Statement

Factorize the polynomial expression: 54x² + 42x³ – 30x⁴

Solution Steps

Step 1: Identify the Common Factor

The first step in factoring a polynomial is to identify the common factor of all the terms. In this case, the common factor is 6x².

• Expression after factoring out the common factor: 6x2(9+7x−5x2)6x2(9+7x−5x2)

Step 2: Factorize the Quadratic Expression

Now, we need to factorize the quadratic expression inside the parentheses. For this, we can use methods like grouping or the quadratic formula.

• Quadratic expression after factoring: 6x2(3−x)(3+5x)6x2(3−x)(3+5x)

Step 3: Final Factorization

Combine the factored common factor with the factored quadratic expression:

• Final factorization of the given polynomial: 6x2(3−x)(3+5x)6x2(3−x)(3+5x)
  
  In conclusion, the factorization of the given polynomial expression is 6x2(3−x)(3+5x)6x2(3−x)(3+5x). When considering Class 7 Tuition, online coaching provides a conducive learning environment with personalized attention, flexibility, abundant resources, technology integration, and expert guidance.

As a seasoned tutor registered on UrbanPro.com, I specialize in providing top-notch online coaching for Class 7 Tuition. Today, I'll guide you through the process of factorizing the expression: 2x²yz + 2xy²z + 4xyz.

Factorization of 2x²yz + 2xy²z + 4xyz

To factorize the given expression, we'll look for common factors in each term and factor them out.

1. Identify Common Factors

   • All terms have a common factor of 2.
   • Each term contains a factor of xyz.

2. Factorize the Expression

   • Factor out the common factor of 2xyz:
     scss

2. • 2xyz(x + y + 2)

Explanation

• Common Factor of 2: Factoring out 2 helps simplify the expression and identify a common factor in each term.

• Common Factor of xyz: Each term contains a factor of xyz. Factoring this out leaves us with the expression (x + y + 2).

• Final Factored Expression: Combining the common factors, the fully factorized expression is 2xyz(x + y + 2).
  
  

  Conclusion

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As an experienced tutor registered on UrbanPro.com, I'll guide you through the process of factoring the quadratic expression 100x² – 80xy + 16y². Let's break down the solution into clear steps.

Step 1: Identify the Type of Quadratic Expression

Before factoring, it's essential to recognize the type of quadratic expression we're dealing with. The given expression is a perfect square trinomial, which can be factored using a specific formula.

Step 2: Apply the Perfect Square Trinomial Formula

The expression 100x² – 80xy + 16y² falls under the category of (a - b)², where 'a' and 'b' are terms in the form of ax and by, respectively. The formula for factoring a perfect square trinomial is:

(a−b)2=a2−2ab+b2(a−b)2=a2−2ab+b2

In our case, a=10xa=10x and b=4yb=4y. Applying the formula:

(10x−4y)2=(10x)2−2(10x)(4y)+(4y)2(10x−4y)2=(10x)2−2(10x)(4y)+(4y)2

Step 3: Simplify the Expression

Now, let's simplify the expression obtained from the formula:

100x2−80xy+16y2100x2−80xy+16y2

=100x2−80xy+16y2=100x2−80xy+16y2

This is the factored form of the given quadratic expression.