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Difficulty: HardFactoring Polynomials

The polynomial x3+5x2−9x−45x^3 + 5x^2 - 9x - 45 can be factored completely into three linear factors of the form (x−a)(x−b)(x−c)(x - a)(x - b)(x - c), where aa, bb, and cc are integers such that a<b<ca < b < c. What is the value of a−b+ca - b + c?

Answer: 1

Answer

The correct answer is 1.
Factoring the polynomial by grouping gives (x2−9)(x+5)(x^2 - 9)(x + 5), which simplifies to (x−3)(x+3)(x+5)(x - 3)(x + 3)(x + 5) after factoring the difference of squares. Writing this expression in the form (x−a)(x−b)(x−c)(x - a)(x - b)(x - c) identifies the values as 33, −3-3, and −5-5. Ordering these values to satisfy the inequality a<b<ca < b < c yields a=−5a = -5, b=−3b = -3, and c=3c = 3. Evaluating a−b+ca - b + c gives −5−(−3)+3=1-5 - (-3) + 3 = 1.

Step-by-Step Solution

1
Group the terms of the polynomial x3+5x2−9x−45x^3 + 5x^2 - 9x - 45.
(x3+5x2)−(9x+45)(x^3 + 5x^2) - (9x + 45)
Grouping allows factoring by finding common binomial terms in a cubic polynomial.
2
Factor out the greatest common factor (GCF) from each grouped term.
x2(x+5)−9(x+5)x^2(x + 5) - 9(x + 5)
The GCF of the first group is x2x^2 and the GCF of the second group is 99.
3
Factor out the common binomial factor (x+5)(x + 5).
(x2−9)(x+5)(x^2 - 9)(x + 5)
Both terms share the common factor (x+5)(x + 5).
4
Factor the quadratic term x2−9x^2 - 9 as a difference of squares.
(x−3)(x+3)(x+5)(x - 3)(x + 3)(x + 5)
x2−9x^2 - 9 is a difference of squares, which factors into (x−3)(x+3)(x - 3)(x + 3).
5
Rewrite the factors in the form (x−a)(x−b)(x−c)(x - a)(x - b)(x - c) to identify the values of the constants.
(x−3)(x−(−3))(x−(−5))(x - 3)(x - (-3))(x - (-5)) which gives the set of values {3,−3,−5}\{3, -3, -5\}.
Matching the signs of the given form (x−constant)(x - \text{constant}) is necessary to correctly identify the values of the constants.
6
Sort the values in ascending order to satisfy a<b<ca < b < c.
a=−5a = -5, b=−3b = -3, and c=3c = 3
The inequality constraint requires sorting the values from smallest to largest.
7
Calculate the value of the expression a−b+ca - b + c.
−5−(−3)+3=1-5 - (-3) + 3 = 1
Substitute the sorted values into the target expression.

Key Concept

Factoring a cubic polynomial by grouping and difference of squares, and identifying algebraic constants under inequality constraints.
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Topics

Factoring Polynomials

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