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Difficulty: MediumFactoring Polynomials

Complete the factored form of the polynomial by finding the missing values.

Answer:The polynomial 6x2−6x−726x^2 - 6x - 72 can be factored completely into the form 6(x−a)(x+b)6(x - a)(x + b). The missing positive integers are a=a = 【4】 and b=b = 【3】.

Answer

The missing values are a = 4 and b = 3. The completely factored form is 6(x - 4)(x + 3).
To factor the polynomial 6x2−6x−726x^2 - 6x - 72 completely, we first factor out the greatest common factor (GCF), which is 66. This leaves us with the expression 6(x2−x−12)6(x^2 - x - 12). Next, we factor the quadratic trinomial x2−x−12x^2 - x - 12. We need to find two integers that multiply to give −12-12 and add up to −1-1. Those integers are −4-4 and 33. This means the trinomial factors as (x−4)(x+3)(x - 4)(x + 3). Combining these, the complete factorization is 6(x−4)(x+3)6(x - 4)(x + 3). By comparing this result to the given format 6(x−a)(x+b)6(x - a)(x + b), we can see that the first missing value is a=4a = 4 and the second missing value is b=3b = 3.

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from the polynomial.
The GCF of the terms 6x26x^2, −6x-6x, and −72-72 is 66. Factoring this out gives: 6(x2−x−12)6(x^2 - x - 12).
Factoring out the GCF simplifies the remaining quadratic expression, making it easier to factor into binomials.
2
Factor the remaining quadratic trinomial x2−x−12x^2 - x - 12.
We look for two numbers that multiply to −12-12 and add up to −1-1 (the coefficient of the middle term). These numbers are −4-4 and 33. Thus, x2−x−12x^2 - x - 12 factors to (x−4)(x+3)(x - 4)(x + 3).
A monic quadratic trinomial x2+px+qx^2 + px + q factors into (x+m)(x+n)(x + m)(x + n) where m×n=qm \times n = q and m+n=pm + n = p.
3
Write the complete factorization and match it to the given format.
The completely factored expression is 6(x−4)(x+3)6(x - 4)(x + 3). Matching this to the given form 6(x−a)(x+b)6(x - a)(x + b), we determine that a=4a = 4 and b=3b = 3.
This directly identifies the specific positive integers required to complete the given algebraic expression.

Key Concept

Factoring out a Greatest Common Factor (GCF) followed by factoring a standard quadratic trinomial.
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Topics

Factoring Polynomials

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