1 1 Linear 2 2 Quadratics

Algebra Level 2

The polynomial 3 x 5 + 17 x 4 + 3 x 3 + 3 x 2 + 17 x + 3 3x^5+17x^4+3x^3+3x^2+17x+3 can be factorized as the product of one linear and two quadratics polynomials, whose leading coefficients are positive. What is the sum of all of the coefficients in the three factors?

26 26 29 29 28 28 27 27

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Mahindra Jain by Mahindra Jain

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4 solutions

3 x 5 + 17 x 4 + 3 x 3 + 3 x 2 + 17 x + 3 3x^5+17x^4+3x^3+3x^2+17x+3

= x 3 ( 3 x 2 + 17 x + 3 ) + 3 x 2 + 17 x + 3 =x^3\left(3x^2+17x+3\right)+3x^2+17x+3

= ( 3 x 2 + 17 x + 3 ) ( x 3 + 1 ) =\left(3x^2+17x+3\right)\left(x^3+1\right)

= ( 3 x 2 + 17 x + 3 ) ( x + 1 ) ( x 2 − x + 1 ) =\left(3x^2+17x+3\right)\left(x+1\right)\left(x^2-x+1\right)

6 Helpful 0 Interesting 0 Brilliant 0 Confused

3 x 5 + 17 x 4 + 3 x 3 + 3 x 2 + 17 x + 3 = 3 x 5 + 3 x 2 + 17 x 4 + 17 x + 3 x 3 + 3 3 x 2 ( x 3 + 1 ) + 17 x ( x 3 + 1 ) + 3 ( x 3 + 1 ) = ( x 3 + 1 ) ( 3 x 2 + 17 x + 3 ) = ( x + 1 ) ( x 2 − x + 1 ) ( 3 x 2 + 17 x + 3 ) \color{#3D99F6}{3x^5+17x^4+3x^3+3x^2+17x+3=3x^5+3x^2+17x^4+17x+3x^3+3\\ 3x^2(x^3+1)+17x(x^3+1)+3(x^3+1)=(x^3+1)(3x^2+17x+3)\\=(x+1)(x^2-x+1)(3x^2+17x+3)} Adding the coefficients we get 26 \boxed{26}

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Dimas Jouhari
Mar 31, 2014

(x+1) (3x^2 + 17x +3) (x^2 - x + 1)

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Aveek Dutta
Mar 22, 2014

(x+1)(x^2 -x +1)(3x^2 + 17x + 3)

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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