Factoring 2x^2 + 5x + 3 Given One Factor (x + 1)

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Factoring polynomials is a fundamental skill in algebra, and understanding how to do it efficiently is crucial for solving various mathematical problems. This article will delve into the process of factoring the quadratic polynomial 2x^2 + 5x + 3, especially when one factor, (x + 1), is already known. We will explore different methods and provide a step-by-step guide to finding the other factor. This comprehensive guide aims to clarify the concepts and provide a solid understanding of polynomial factorization.

Understanding Polynomial Factorization

In the realm of algebra, a polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Factoring a polynomial involves breaking it down into simpler expressions (factors) that, when multiplied together, yield the original polynomial. This process is akin to finding the prime factors of an integer, but applied to algebraic expressions. Polynomial factorization is essential for solving equations, simplifying expressions, and analyzing mathematical functions.

The polynomial 2x^2 + 5x + 3 is a quadratic polynomial, which means it has a degree of 2 (the highest power of the variable x is 2). Quadratic polynomials have the general form ax^2 + bx + c, where a, b, and c are constants. Factoring a quadratic polynomial involves finding two binomials (expressions with two terms) that multiply to give the original quadratic. Understanding this foundational concept is crucial before diving into the specifics of factoring 2x^2 + 5x + 3.

The significance of factoring polynomials extends beyond mere algebraic manipulation. It is a cornerstone of solving quadratic equations, which arise in various real-world applications, including physics, engineering, and economics. For example, factoring can help determine the trajectory of a projectile, the optimal dimensions of a structure, or the equilibrium points in a market model. Mastering polynomial factorization, particularly for quadratics, equips individuals with a powerful tool for problem-solving and mathematical analysis. This skill not only enhances algebraic proficiency but also fosters critical thinking and logical reasoning, essential for navigating diverse academic and professional challenges.

Methods to Factor 2x^2 + 5x + 3

When faced with the task of factoring a polynomial, several methods can be employed, each with its own strengths and applications. For the quadratic polynomial 2x^2 + 5x + 3, we can consider the following approaches:

1. Trial and Error

This method involves systematically trying different combinations of binomial factors until the correct one is found. It relies on an understanding of how binomials multiply and the distribution property. For 2x^2 + 5x + 3, we know that the factors will have the form (ax + b) and (cx + d), where a, b, c, and d are constants. The product of a and c must equal 2 (the coefficient of x^2), and the product of b and d must equal 3 (the constant term). The middle term, 5x, arises from the sum of the cross-products (adx + bcx).

Starting with the factors of 2 (1 and 2) and the factors of 3 (1 and 3), we can test different combinations. One possible combination is (2x + 1)(x + 3), which expands to 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3, which is not the original polynomial. Another combination is (2x + 3)(x + 1), which expands to 2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3, matching the original polynomial. Thus, the correct factors are (2x + 3) and (x + 1).

While trial and error can be effective, it may not be the most efficient method for more complex polynomials or when the coefficients are larger. It requires a certain amount of intuition and careful checking of each combination.

2. The AC Method

The AC method is a more systematic approach to factoring quadratic polynomials. It involves the following steps:

  1. Multiply the leading coefficient (a) by the constant term (c). In this case, a = 2 and c = 3, so ac = 2 * 3 = 6.
  2. Find two numbers that multiply to ac (6) and add up to the middle coefficient (b), which is 5. The numbers 2 and 3 satisfy these conditions (2 * 3 = 6 and 2 + 3 = 5).
  3. Rewrite the middle term (5x) using these two numbers: 2x^2 + 5x + 3 becomes 2x^2 + 2x + 3x + 3.
  4. Factor by grouping. Group the first two terms and the last two terms: (2x^2 + 2x) + (3x + 3).
  5. Factor out the greatest common factor (GCF) from each group: 2x(x + 1) + 3(x + 1).
  6. Notice that (x + 1) is a common factor. Factor it out: (x + 1)(2x + 3).

The AC method provides a clear and structured approach to factoring quadratics, reducing the guesswork involved in the trial and error method.

3. Using the Given Factor (x + 1)

Since we are given that (x + 1) is one factor of 2x^2 + 5x + 3, we can use this information to find the other factor more directly. This can be done through polynomial long division or synthetic division.

Polynomial Long Division

Polynomial long division is analogous to long division with numbers. We divide 2x^2 + 5x + 3 by (x + 1):

 2x + 3
x + 1 | 2x^2 + 5x + 3
 - (2x^2 + 2x)
 ----------------
 3x + 3
 - (3x + 3)
 ----------------
 0

The quotient, 2x + 3, is the other factor. The remainder is 0, confirming that (x + 1) is indeed a factor.

Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear factor of the form (x - c). In this case, we divide by (x + 1), so c = -1.

  1. Write down the coefficients of the polynomial: 2, 5, and 3.
  2. Write -1 (the value of c) to the left.
  3. Perform the synthetic division:
 -1 | 2 5 3
 | -2 -3
 ----------------
 2 3 0

The numbers 2 and 3 are the coefficients of the quotient, which is 2x + 3. The 0 represents the remainder, again confirming that (x + 1) is a factor.

Using the given factor simplifies the factoring process significantly, as it narrows down the possibilities and allows for a more direct route to the solution.

Step-by-Step Solution Using the Given Factor

Given that (x + 1) is one factor of the polynomial 2x^2 + 5x + 3, we can efficiently find the other factor using either polynomial long division or synthetic division. Both methods leverage the information provided to streamline the factorization process.

1. Polynomial Long Division Method

Polynomial long division is a systematic way to divide polynomials, similar to long division with numbers. Here's how we can apply it to find the other factor:

  1. Set up the division: Place the polynomial 2x^2 + 5x + 3 inside the division symbol and the given factor (x + 1) outside.
     ________
x + 1 | 2x^2 + 5x + 3
  1. Divide the first term of the dividend (2x^2) by the first term of the divisor (x). This gives us 2x. Write 2x above the division symbol.
     2x    
x + 1 | 2x^2 + 5x + 3
  1. Multiply the divisor (x + 1) by 2x: 2x(x + 1) = 2x^2 + 2x. Write this result below the corresponding terms in the dividend.
     2x    
x + 1 | 2x^2 + 5x + 3
      2x^2 + 2x
  1. Subtract the result from the dividend: (2x^2 + 5x + 3) - (2x^2 + 2x) = 3x + 3. Write this below.
     2x    
x + 1 | 2x^2 + 5x + 3
      2x^2 + 2x
      -------
      3x + 3
  1. Divide the first term of the new dividend (3x) by the first term of the divisor (x). This gives us 3. Write +3 next to 2x above the division symbol.
     2x + 3
x + 1 | 2x^2 + 5x + 3
      2x^2 + 2x
      -------
      3x + 3
  1. Multiply the divisor (x + 1) by 3: 3(x + 1) = 3x + 3. Write this below the current remainder.
     2x + 3
x + 1 | 2x^2 + 5x + 3
      2x^2 + 2x
      -------
      3x + 3
      3x + 3
  1. Subtract the result from the remainder: (3x + 3) - (3x + 3) = 0. This confirms that (x + 1) is indeed a factor and that 2x + 3 is the other factor.
     2x + 3
x + 1 | 2x^2 + 5x + 3
      2x^2 + 2x
      -------
      3x + 3
      3x + 3
      -------
      0

Therefore, using polynomial long division, we find that the other factor is 2x + 3.

2. Synthetic Division Method

Synthetic division is a more streamlined method for dividing a polynomial by a linear factor of the form (x - c). In our case, we are dividing by (x + 1), so c = -1. Here’s the step-by-step process:

  1. Write down the coefficients of the polynomial 2x^2 + 5x + 3: 2, 5, and 3. Also, write the value of c (-1) to the left.
-1 | 2 5 3
   ----------------
  1. Bring down the first coefficient (2) below the line.
-1 | 2 5 3
   ----------------
    2
  1. Multiply the number you brought down (2) by c (-1): 2 * (-1) = -2. Write this result below the second coefficient (5).
-1 | 2 5 3
    -2
   ----------------
    2
  1. Add the second coefficient (5) and the result (-2): 5 + (-2) = 3. Write this below the line.
-1 | 2 5 3
    -2
   ----------------
    2 3
  1. Multiply the new number below the line (3) by c (-1): 3 * (-1) = -3. Write this result below the third coefficient (3).
-1 | 2 5 3
    -2 -3
   ----------------
    2 3
  1. Add the third coefficient (3) and the result (-3): 3 + (-3) = 0. Write this below the line. The 0 represents the remainder.
-1 | 2 5 3
    -2 -3
   ----------------
    2 3 0
  1. Interpret the numbers below the line. The numbers 2 and 3 are the coefficients of the quotient, which is 2x + 3. The 0 represents the remainder, confirming that (x + 1) is a factor.

Using synthetic division, we also find that the other factor is 2x + 3.

Both polynomial long division and synthetic division provide efficient methods to find the other factor when one factor is known. These techniques are invaluable tools in polynomial factorization, especially in more complex scenarios. The step-by-step application of these methods ensures accuracy and clarity in the solution process.

Conclusion

In conclusion, factoring the polynomial 2x^2 + 5x + 3, given that one factor is (x + 1), leads us to the other factor, which is 2x + 3. We explored multiple methods, including trial and error, the AC method, polynomial long division, and synthetic division. The most efficient methods, polynomial long division and synthetic division, directly utilize the known factor to simplify the process. These methods not only provide the solution but also reinforce the understanding of polynomial factorization techniques.

The ability to factor polynomials is a fundamental skill in algebra, with applications spanning various mathematical and scientific disciplines. Understanding and practicing these methods enhances problem-solving skills and provides a solid foundation for more advanced mathematical concepts. Whether you choose the systematic approach of the AC method, the visual clarity of polynomial long division, or the streamlined efficiency of synthetic division, mastering polynomial factorization equips you with a powerful tool for mathematical analysis and problem-solving. The result 2x + 3 is crucial for solving related algebraic problems and comprehending the underlying principles of polynomial manipulation.

Therefore, the correct answer is B. 2x + 3.