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Difficulty: Very hardFactoring Polynomials

If the expression x4−8x2+16−9y2x^4 - 8x^2 + 16 - 9y^2 is factored completely over the integers, the product of the factors can be written as (x2−a−by)(x2−c+dy)(x^2 - a - by)(x^2 - c + dy), where aa, bb, cc, and dd are positive integers. What is the value of a+b+c+da + b + c + d?

Answer: 14

Answer

14
By grouping the first three terms, the expression x4−8x2+16x^4 - 8x^2 + 16 is recognized as (x2−4)2(x^2 - 4)^2. Substituting this back into the original expression gives (x2−4)2−(3y)2(x^2 - 4)^2 - (3y)^2. Applying the difference of squares identity, this factors into (x2−4−3y)(x2−4+3y)(x^2 - 4 - 3y)(x^2 - 4 + 3y). Comparing this result to (x2−a−by)(x2−c+dy)(x^2 - a - by)(x^2 - c + dy) where a,b,c,da, b, c, d are positive integers yields a=4a = 4, b=3b = 3, c=4c = 4, and d=3d = 3. Summing these values gives 4+3+4+3=144 + 3 + 4 + 3 = 14.

Step-by-Step Solution

1
Group the first three terms of the polynomial.
x4−8x2+16=(x2−4)2x^4 - 8x^2 + 16 = (x^2 - 4)^2
To recognize the perfect square trinomial structure in terms of x2x^2.
2
Rewrite the original expression using the grouped terms.
(x2−4)2−9y2=(x2−4)2−(3y)2(x^2 - 4)^2 - 9y^2 = (x^2 - 4)^2 - (3y)^2
To express the polynomial as a difference of squares.
3
Factor the expression using the difference of squares formula A2−B2=(A−B)(A+B)A^2 - B^2 = (A - B)(A + B).
(x2−4−3y)(x2−4+3y)(x^2 - 4 - 3y)(x^2 - 4 + 3y)
To obtain the completely factored form over the integers.
4
Compare the factored expression to the given template (x2−a−by)(x2−c+dy)(x^2 - a - by)(x^2 - c + dy) where a,b,c,da, b, c, d are positive integers.
a=4a = 4, b=3b = 3, c=4c = 4, d=3d = 3
To identify the values of the constants that satisfy the positivity constraint.
5
Calculate the sum of the identified values.
a+b+c+d=4+3+4+3=14a + b + c + d = 4 + 3 + 4 + 3 = 14
To answer the question.

Key Concept

Factoring by grouping and the difference of squares
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Topics

Factoring Polynomials

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