The coefficient of x2 in 3x3+2x2-x+1 is

General
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the coefficient of x2 in 3x3+2x2-x+1 is

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What is the coefficient of x^2 in 3x^3 + 2x^2 - x + 1?

Identifying the Coefficient:

In a polynomial, each term is made up of a coefficient and a variable raised to a power. The coefficient is the numerical factor that multiplies the variable part of the term. To find the coefficient of x^2 in a polynomial, you need to identify the term that contains x^2 and then locate its coefficient.

In the polynomial 3x^3 + 2x^2 - x + 1, the term that contains x^2 is 2x^2. Here, the number directly in front of x^2 is the coefficient. Therefore, the coefficient of x^2 is 2.

Step-by-Step Explanation:

  1. Identify Terms: A polynomial is a sum of terms, each consisting of a coefficient and a variable raised to a power. The polynomial given is 3x^3 + 2x^2 - x + 1.

  2. Look for x^2: We are interested in finding the term with x^2.

  3. Focus on the Correct Term: In the polynomial 3x^3 + 2x^2 - x + 1, the term containing x^2 is 2x^2.

  4. Check the Coefficient: The coefficient of x^2 in 2x^2 is 2.

To reinforce understanding, let’s consider another example and additional exercises.

Example: Find the Coefficient of x^4 in the Polynomial 5x^4 - 3x^3 + 7x^2 + 10x - 6.

  1. Identify the Terms: Here, the terms are 5x^4, -3x^3, 7x^2, 10x, and -6.

  2. Locate the x^4 Term: The term in this polynomial that contains x^4 is 5x^4.

  3. Determine the Coefficient: The coefficient of x^4 is 5.

Thus, the coefficient of x^4 in this polynomial is 5.

Practice Exercises:

Let’s practice with more polynomials to solidify the concept. Find the coefficient of the indicated term in each polynomial.

  • Polynomial: 4x^5 + 6x^4 - 2x^3 + 8x - 1, Find the Coefficient of x^3.

    • Solution: The term with x^3 is -2x^3, so the coefficient is -2.

  • Polynomial: 7x^6 + 5x^4 + 9x^2 + 3, Find the Coefficient of x^4.

    • Solution: The term with x^4 is 5x^4, so the coefficient is 5.

  • Polynomial: x^8 - 11x^7 + x^5 - x^2 + 9, Find the Coefficient of x^2.

    • Solution: The term with x^2 is -x^2, and the coefficient is -1.

  • Polynomial: 12x^9 - 4x^3 + 17x + 5, Find the Coefficient of x^3.

    • Solution: The term with x^3 is -4x^3, so the coefficient is -4.

Real-World Analogy:

Understanding coefficients can be compared to shopping for produce. Imagine each term of a polynomial as a different fruit basket. Each basket has a certain number of fruits (coefficients) of a specific type (variable and its power).

  • The term 2x^2 is like a basket with 2 apples (where apple is x^2).
  • Identifying the coefficient is like counting how many apples are in the basket, which, in this case, is 2.

Summary:

The coefficient of x^2 in the polynomial 3x^3 + 2x^2 - x + 1 is 2. Identifying coefficients involves recognizing the numerical part of a term that multiplies the variable component. Practicing with a variety of polynomials helps strengthen this foundational algebraic skill.

Feel free to let me know if you need more assistance or examples, @anonymous6!