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Difficulty: MediumFactoring Polynomials

If the polynomial x4−13x2+36x^4 - 13x^2 + 36 is factored completely into the product of four linear binomials of the form (x−r1)(x−r2)(x−r3)(x−r4)(x - r_1)(x - r_2)(x - r_3)(x - r_4), where r1<r2<r3<r4r_1 < r_2 < r_3 < r_4, what is the value of the expression r1+2r2+3r3+4r4r_1 + 2r_2 + 3r_3 + 4r_4?

Answer: 11

Answer

The value of the expression is 11.
Factoring the quartic polynomial x4−13x2+36x^4 - 13x^2 + 36 as a quadratic in x2x^2 yields (x2−9)(x2−4)(x^2 - 9)(x^2 - 4). Applying the difference of squares identity to each factor gives (x−3)(x+3)(x−2)(x+2)(x - 3)(x + 3)(x - 2)(x + 2). Matching these to the template (x−r1)(x−r2)(x−r3)(x−r4)(x - r_1)(x - r_2)(x - r_3)(x - r_4) with the condition r1<r2<r3<r4r_1 < r_2 < r_3 < r_4 yields r1=−3r_1 = -3, r2=−2r_2 = -2, r3=2r_3 = 2, and r4=3r_4 = 3. Evaluating the linear combination −3+2(−2)+3(2)+4(3)-3 + 2(-2) + 3(2) + 4(3) results in 1111.

Step-by-Step Solution

1
Substitute u=x2u = x^2 and factor the quadratic trinomial.
u2−13u+36=(u−9)(u−4)⇒(x2−9)(x2−4)u^2 - 13u + 36 = (u - 9)(u - 4) \Rightarrow (x^2 - 9)(x^2 - 4)
To reduce the degree of the polynomial and make it easier to factor.
2
Apply the difference of squares formula to each binomial factor.
(x−3)(x+3)(x−2)(x+2)(x - 3)(x + 3)(x - 2)(x + 2)
Both x2−9x^2 - 9 and x2−4x^2 - 4 are differences of squares.
3
Determine the roots r1,r2,r3,r4r_1, r_2, r_3, r_4 in ascending order.
r1=−3r_1 = -3, r2=−2r_2 = -2, r3=2r_3 = 2, r4=3r_4 = 3
We write each factor as (x−ri)(x - r_i) to find the roots, and then sort them from least to greatest according to the inequality constraint.
4
Calculate the value of the requested expression.
1111
Substitute the sorted values into the linear combination: (−3)+2(−2)+3(2)+4(3)=−3−4+6+12=11(-3) + 2(-2) + 3(2) + 4(3) = -3 - 4 + 6 + 12 = 11.

Key Concept

Factoring a quartic polynomial of quadratic form followed by factoring differences of squares.
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Topics

Factoring Polynomials

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