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Question

Difficulty: HardSimplifying and Factoring Algebraic Expressions
For all real numbers xx such that x≠3x \neq 3 and x≠−5x \neq -5, the algebraic expression
(x2−9)2−4(x−3)2(x−3)(x+5)\frac{(x^2 - 9)^2 - 4(x - 3)^2}{(x - 3)(x + 5)}
can be simplified to the equivalent polynomial expression x2+ax+bx^2 + ax + b, where aa and bb are constants. What is the value of a+ba + b?

Answer: -5

Answer

The value of a+ba + b is −5-5.
Factoring the numerator yields (x−3)2(x+1)(x+5)(x - 3)^2(x + 1)(x + 5). Canceling the common factors (x−3)(x - 3) and (x+5)(x + 5) with the denominator simplifies the expression to (x−3)(x+1)=x2−2x−3(x - 3)(x + 1) = x^2 - 2x - 3. Comparing this to x2+ax+bx^2 + ax + b identifies a=−2a = -2 and b=−3b = -3, giving a sum of a+b=−5a + b = -5.

Step-by-Step Solution

1
Factor the numerator by recognizing (x2−9)=(x−3)(x+3)(x^2 - 9) = (x - 3)(x + 3)
(x2−9)2−4(x−3)2=[(x−3)(x+3)]2−4(x−3)2=(x−3)2(x+3)2−4(x−3)2(x^2 - 9)^2 - 4(x - 3)^2 = [(x - 3)(x + 3)]^2 - 4(x - 3)^2 = (x - 3)^2 (x + 3)^2 - 4(x - 3)^2
Applying the difference of squares identity inside the squared term allows factoring out common factors.
2
Factor out (x−3)2(x - 3)^2 from the numerator
(x−3)2[(x+3)2−4](x - 3)^2 \left[ (x + 3)^2 - 4 \right]
Extracting the greatest common algebraic factor simplifies the remaining expression.
3
Apply difference of squares to (x+3)2−4(x + 3)^2 - 4
(x+3)2−22=((x+3)−2)((x+3)+2)=(x+1)(x+5)(x + 3)^2 - 2^2 = ((x + 3) - 2)((x + 3) + 2) = (x + 1)(x + 5)
Recognizing (x+3)2−22(x + 3)^2 - 2^2 as A2−B2A^2 - B^2 yields factored linear terms directly.
4
Substitute the fully factored numerator back into the rational expression and simplify
\frac{(x - 3)^2 (x + 1)(x + 5)}{(x - 3)(x + 5)} = (x - 3)(x + 1)
Canceling non-zero common factors (x−3)(x - 3) and (x+5)(x + 5) simplifies the rational function.
5
Expand (x−3)(x+1)(x - 3)(x + 1) and determine a+ba + b
(x−3)(x+1)=x2−2x−3(x - 3)(x + 1) = x^2 - 2x - 3, so a=−2a = -2 and b=−3b = -3. Therefore, a+b=−2+(−3)=−5a + b = -2 + (-3) = -5.
Matching coefficients with x2+ax+bx^2 + ax + b gives a=−2a = -2 and b=−3b = -3.

Key Concept

Simplifying complex rational expressions through nested difference of squares factoring.
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Topics

Simplifying and Factoring Algebraic Expressions

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