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Zorluk: ZorFactoring Polynomials

The cubic polynomial 6x3−19x2+11x+66x^3 - 19x^2 + 11x + 6 can be factored completely over the integers in the form (x−a)(bx−c)(dx+e)(x - a)(bx - c)(dx + e), where aa, bb, cc, dd, and ee are positive integers. What is the value of a+b+c+d+ea + b + c + d + e?

Cevap: 11

Cevap

The sum of the coefficients and constants from the factored form is 11.
By applying the Rational Root Theorem, we find the root x=2x = 2, which gives the factor (x−2)(x - 2). Dividing the original cubic expression by (x−2)(x - 2) yields 6x2−7x−36x^2 - 7x - 3. Factoring this quadratic expression by grouping yields (2x−3)(3x+1)(2x - 3)(3x + 1). Writing the completely factored form as (x−2)(2x−3)(3x+1)(x - 2)(2x - 3)(3x + 1) and comparing it to (x−a)(bx−c)(dx+e)(x - a)(bx - c)(dx + e) where a,b,c,d,ea, b, c, d, e are positive integers results in a=2a = 2, b=2b = 2, c=3c = 3, d=3d = 3, and e=1e = 1. The sum a+b+c+d+ea + b + c + d + e is equal to 11.

Adım Adım Çözüm

1
Identify a rational root of 6x3−19x2+11x+66x^3 - 19x^2 + 11x + 6
x=2x = 2 is a root, meaning (x−2)(x - 2) is a factor.
Applying the Rational Root Theorem and testing potential integer root values.
2
Divide 6x3−19x2+11x+66x^3 - 19x^2 + 11x + 6 by (x−2)(x - 2)
The quotient is the quadratic polynomial 6x2−7x−36x^2 - 7x - 3.
To reduce the cubic polynomial to a quadratic expression that can be factored using standard trinomial methods.
3
Factor the quadratic trinomial 6x2−7x−36x^2 - 7x - 3
(2x−3)(3x+1)(2x - 3)(3x + 1)
Finding two numbers that multiply to −18-18 and add to −7-7 (which are −9-9 and 22) and factoring by grouping.
4
Match the factored form (x−2)(2x−3)(3x+1)(x - 2)(2x - 3)(3x + 1) to (x−a)(bx−c)(dx+e)(x - a)(bx - c)(dx + e)
a=2,b=2,c=3,d=3,e=1a = 2, b = 2, c = 3, d = 3, e = 1
Since a,b,c,d,a, b, c, d, and ee must be positive integers, the constant terms and signs uniquely determine the mapping of each factor.
5
Calculate the sum of a,b,c,d,a, b, c, d, and ee
2+2+3+3+1=112 + 2 + 3 + 3 + 1 = 11
To find the final numeric value requested by the question.

Anahtar Kavram

Factoring cubic polynomials by finding rational roots and factoring quadratic trinomials by grouping.
Tahmini Süre:2m 30s
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Factoring Polynomials

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