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Chapter 5: Problem 85

Factor each polynomial completely. $$ -9 x^{2}+33 x+12 $$

Short Answer

Expert verified
\[ 3(3x + 1)(-x + 4) \].

Step by step solution

01

- Identify a GCF (Greatest Common Factor)

Check if the polynomial has a GCF other than 1. In this case, the coefficients are -9, 33, and 12. The GCF of these numbers is 3, which we factor out: \[ -9x^2 + 33x + 12 = 3(-3x^2 + 11x + 4) \].
02

- Rewrite the Polynomial

Now, focus on the quadratic polynomial inside the parentheses: \[ -3x^2 + 11x + 4 \].
03

- Factorize the Quadratic Polynomial

Look for two numbers that multiply to the product of -3 and 4 (which is -12) and add to 11. These numbers are -1 and 12. We rewrite the middle term using these two numbers: \[ -3x^2 - x + 12x + 4 \].
04

- Group Terms

Group the polynomial into two binomials: \[ (-3x^2 - x) + (12x + 4) \].
05

- Factor Each Group

Factor out the GCF from each group: \[ -x(3x + 1) + 4(3x + 1) \].
06

- Factor by Grouping

Now factor out the common binomial factor \[ (3x + 1) \]: \[ -x(3x + 1) + 4(3x + 1) = (3x + 1)(-x + 4) \].
07

- Combine all factors

Combine the factors including the GCF from Step 1: \[ 3(3x + 1)(-x + 4) \].

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Key Concepts

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

Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is a fundamental concept in algebra and factoring. It represents the largest number that divides all the coefficients of the terms in a polynomial. Finding the GCF helps simplify polynomials and makes factoring more manageable.

For the polynomial -9x^2 + 33x + 12, we find the GCF by examining the coefficients: -9, 33, and 12. The largest number that divides all three is 3. Therefore, we factor out 3:
-9x^2 + 33x + 12 = 3(-3x^2 + 11x + 4) .

This step simplifies the polynomial, making further factoring easier.
Quadratic Polynomial
A quadratic polynomial is a second-degree polynomial with the general form ax^2 + bx + c. The polynomial inside the parentheses (-3x^2 + 11x + 4) is a quadratic polynomial. We need to factor it to solve the problem.

To factor this polynomial, look for two numbers that multiply to the product of the leading coefficient (-3) and the constant term (4), which gives us -12. These two numbers also need to add up to the middle coefficient (11). After some consideration, we find -1 and 12 meet these criteria.
Factoring by Grouping
Factoring by grouping is a technique used to factor polynomials with four terms. It involves grouping terms in pairs and factoring out the GCF from each pair.

Let's continue from our previous example:
The quadratic polynomial -3x^2 + 11x + 4 is rewritten using the numbers we found:
-3x^2 - x + 12x + 4 .

Next, group the terms:
(-3x^2 - x) + (12x + 4) .

Factor each group:
-x(3x + 1) + 4(3x + 1) .

Notice that (3x + 1) is a common factor:
(3x + 1)(-x + 4) .

Finally, include the GCF we factored out initially:
3(3x + 1)(-x + 4) .

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