Math Problem Statement

Factor the given expression completely: 7z^2 - 15z + 2

Solution

The given expression is:

7z215z+27z^2 - 15z + 2

To factor this, let’s proceed step by step:

Step 1: Multiply the leading coefficient and the constant term

The leading coefficient is 77, and the constant term is 22. Their product is:

7×2=147 \times 2 = 14

Step 2: Find two numbers that multiply to 1414 and add to 15-15

The two numbers are 14-14 and 1-1, since:

14×1=14and14+1=15-14 \times -1 = 14 \quad \text{and} \quad -14 + -1 = -15

Step 3: Rewrite the middle term using these numbers

Rewrite 15z-15z as 14zz-14z - z:

7z214zz+27z^2 - 14z - z + 2

Step 4: Group and factor each group

Group the terms into two pairs:

(7z214z)(z2)(7z^2 - 14z) - (z - 2)

Factor out the greatest common factor (GCF) from each group:

7z(z2)1(z2)7z(z - 2) - 1(z - 2)

Step 5: Factor out the common binomial factor

(7z1)(z2)(7z - 1)(z - 2)

Final Answer:

The completely factored form of 7z215z+27z^2 - 15z + 2 is:

(7z1)(z2)(7z - 1)(z - 2)

Let me know if you’d like further explanation or help with related questions!


5 Related Questions:

  1. How do you check if a quadratic expression is factorable or prime?
  2. What is the process for factoring when the leading coefficient is greater than 11?
  3. How do you determine the greatest common factor (GCF) in a polynomial?
  4. What is the role of the discriminant in checking factorability of quadratic equations?
  5. Can all quadratic expressions be factored over integers?

Tip:

Always check your factors by expanding them back to the original expression to ensure correctness!

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring Quadratics
Grouping Method

Formulas

Standard quadratic form: ax^2 + bx + c
Factoring method: Find two numbers that multiply to ac and add to b

Theorems

Distributive Property

Suitable Grade Level

Grades 9-11