Answered

Which is the fully factored form of the polynomial:

$4 a^3+2 a^2 b-30 a b^2$


A$2 a\left(2 a^2-a b-2 b^2\right)$
B$2 a(a+3 b)(2 a-5 b)$
C$(2 a+6 a b)\left(2 a^2-5 b\right)$
D$2\left(2 a^2-3 b\right)(a+10 b)$

Verified Answer :

  • Extract the greatest common factor \(2a\) from the polynomial: \(2a(2a^2 + ab - 15b^2)\).
  • Factor the remaining quadratic trinomial \(2a^2 + ab - 15b^2\) into two binomials \((2a - 5b)(a + 3b)\).
  • Assemble the factors to get the full expression: \(2a(a + 3b)(2a - 5b)\).
  • Identify the correct option, which is \(\boxed{B}\).

Explanation

  1. Problem Analysis and Initial Factoring

We are given the polynomial \(P(a, b) = 4 a^3 + 2 a^2 b - 30 a b^2\). Our objective is to factor this expression completely to find which of the provided options matches it. Factoring a polynomial involves breaking it down into a product of simpler polynomials. We start by looking for the greatest common factor (GCF) that can be extracted from all the terms.

  1. Factoring the Quadratic Expression

Looking at the terms \(4a^3\), \(2a^2b\), and \(-30ab^2\), we see that each term contains at least one factor of \(a\) and each coefficient is divisible by \(2\). Therefore, we can factor out the greatest common factor, which is \(2a\):
\[4 a^3 + 2 a^2 b - 30 a b^2 = 2a(2a^2 + ab - 15b^2)\]
Now, we need to focus on factoring the remaining quadratic expression inside the parentheses: \(2a^2 + ab - 15b^2\).

  1. Final Factoring and Comparison

To factor the quadratic \(2a^2 + ab - 15b^2\), we look for two binomial factors of the form \((xa + yb)(za + wb)\). We know that:

  1. The product of the first terms must be \(2a^2\), so we can set \(x=2\) and \(z=1\), giving \((2a + yb)(a + wb)\).
  2. The product of the last terms must be \(-15b^2\).
  3. The sum of the cross terms must be \(+ab\).

Let's test potential factors of \(-15\), such as \((3, -5)\) or \((-3, 5)\).
Testing \((2a - 5b)(a + 3b)\):
\[(2a)(a) + (2a)(3b) + (-5b)(a) + (-5b)(3b) = 2a^2 + 6ab - 5ab - 15b^2 = 2a^2 + ab - 15b^2\]
This expansion matches our quadratic exactly. Therefore, \(2a^2 + ab - 15b^2 = (2a - 5b)(a + 3b)\).

  1. Conclusion

Combining all parts, the fully factored form of the polynomial is:
\[2a(2a^2 + ab - 15b^2) = 2a(2a - 5b)(a + 3b)\]
Comparing this result with the provided options:

A\(2 a(2 a^2 - a b - 2 b^2)\)
B\(2 a(a + 3 b)(2 a - 5 b)\)
C\((2 a + 6 a b)(2 a^2 - 5 b)\)
D\(2(2 a^2 - 3 b)(a + 10 b)\)

The expression matches option B. Thus, the fully factored form is \(2a(a + 3b)(2a - 5b)\). The correct option is B.

Examples

Factoring polynomials is a fundamental skill in algebra with many real-world applications. For example, if you are an engineer designing a rectangular container where the volume is given by a polynomial, you can use factoring to determine the length, width, and height of the container in terms of the variable dimensions. This process simplifies complex models into manageable parts, allowing for easier calculation of material usage, surface area, or structural integrity. By understanding the building blocks of a mathematical expression, you can better analyze and optimize designs in fields ranging from construction to software algorithm optimization.

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