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Chapter 5: Problem 51

Write an equivalent expression by factoring. $$12 a^{4}-21 a^{3}-9 a^{2}$$

Short Answer

Expert verified
3a^2(4a^2 - 7a - 3)

Step by step solution

01

Identify the common factor

Look for the greatest common factor (GCF) in the terms of the given expression. The terms are: 1) 12a^4 2) -21a^3 3) -9a^2. The common factor in these terms is 3a^2.
02

Factor out the common factor

Divide each term by the GCF (3a^2) and write the expression: 12a^4 梅 3a^2 = 4a^2 -21a^3 梅 3a^2 = -7a -9a^2 梅 3a^2 = -3. So, we can factor out 3a^2 from the original expression: 3a^2(4a^2 - 7a - 3)
03

Verify the factored form

Make sure the factored expression, 3a^2(4a^2 - 7a - 3), is equivalent to the original expression. Multiply the factors to verify: (3a^2)(4a^2 - 7a - 3) = 3a^2 * 4a^2 - 3a^2 * 7a - 3a^2 * 3 = 12a^4 - 21a^3 - 9a^2 which matches the original.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

greatest common factor
When factoring polynomial expressions, identifying the greatest common factor (GCF) is the first crucial step. The GCF is the largest factor that divides each term of the polynomial. For the polynomial given in the exercise, 12a^4 - 21a^3 - 9a^2, we start by breaking down each term:
  • 12a^4 can be factored into 2 * 2 * 3 * a * a * a * a
  • -21a^3 can be factored into -1 * 3 * 7 * a * a * a
  • -9a^2 can be factored into -1 * 3 * 3 * a * a
Next, to find the GCF, we look for the highest power of common factors. Here, the common factor is 3a^2 since it appears in all terms. Once the GCF (3a^2) is identified, we factor it out from each term in the polynomial.
polynomial division
After identifying the greatest common factor, the next step is polynomial division. This means dividing each term of the polynomial by the GCF to find what's left inside the parentheses. Let's break it down for the given problem: Starting with the polynomial 12a^4 - 21a^3 - 9a^2:
  • Divide 12a^4 by 3a^2 to get 4a^2.
  • Divide -21a^3 by 3a^2 to get -7a.
  • Divide -9a^2 by 3a^2 to get -3.
This division simplifies the polynomial to 4a^2 - 7a - 3. Now we can write the factored form of the polynomial as 3a^2(4a^2 - 7a - 3) showing that the GCF times the simplified polynomial equals the original expression.
factored form verification
Lastly, it is important to verify the factored form of the polynomial to ensure accuracy. Factored form verification involves multiplying the factors back together to see if they match the original polynomial. For the problem at hand, the factored form is 3a^2(4a^2 - 7a - 3).
To verify:
  • Multiply 3a^2 by each term inside the parentheses:
  • (3a^2 * 4a^2) gives 12a^4.
  • (3a^2 * -7a) gives -21a^3.
  • (3a^2 * -3) gives -9a^2.
Combining these results, the expanded expression is 12a^4 - 21a^3 - 9a^2, which matches the original polynomial. This confirms that our factored expression, 3a^2(4a^2 - 7a - 3), is indeed correct. Verification is a key step because it ensures that the factorization process was done accurately and therefore, can be trusted.

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