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Dividing a Polynomial by a Monomial

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Question

how to solve the equations - 2x² + 7xy + 3y².
x³ + 7x²y + 16xy² + 12y³
3x³ - 14x² + 9x + 10
15x³ - 34x² + 21x - 10

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Solution

1)
$2 x^{2} + 7 x y + 3 y^{2} = 2 x^{2} + 6 x y + x y + 3 y^{2} = 2 x (x + 3 y) + y (x + 3 y) = (x + 3 y) (2 x + y)$

2)
$x^{3} + 7 x^{2} y + 16 x y^{2} + 12 y^{3} B y t r i a l a n d e r r o r m e t h o d w e f o u n d t h a t w h e n x = 3, y = - 1 a n d w h e n x = - 3, y = 1 \Rightarrow x = - 3 y \Rightarrow (x + 3 y) i s a f a c t o r o f x^{3} + 7 x^{2} y + 16 x y^{2} + 12 y^{3} U \sin g l o n g d i v i s i o n m e t h o d w e g e t \sqrt[x + 3 y]{x^{3} + 7 x^{2} y + 16 x y^{2} + 12 y^{3}} x^{2} + 4 x y + 4 y^{2} - x^{3} - 3 x^{2} y 0 + 4 x^{2} y + 16 x y^{2} + 12 y^{3} - 4 x^{2} y - 12 x y^{2} 0 + 4 x y^{2} + 12 y^{3} - 4 x y^{2} - 12 y^{3} 0 \Rightarrow x^{3} + 7 x^{2} y + 16 x y^{2} + 12 y^{3} = (x + 3 y) (x^{2} + 4 x y + 4 y^{2}) = (x + 3 y) (x^{2} + 2 x y + 2 x y + 4 y^{2}) = (x + 3 y) (x (x + 2 y) + 2 y (x + 2 y)) = (x + 3 y) (x + 2 y) (x + 2 y)$

3)
$3 x^{3} - 14 x^{2} + 9 x + 10 U \sin g t r i a l a n d e r r o r m e t h o d w e g e t t h a t 3 x^{3} - 14 x^{2} + 9 x + 10 = 0 w h e n x = \frac{5}{3} \Rightarrow (3 x - 5) i s a f a c t o r o f 3 x^{3} - 14 x^{2} + 9 x + 10 U \sin g l o n g d i v i s i o n w e g e t \sqrt[3 x - 5]{3 x^{3} - 14 x^{2} + 9 x + 10} x^{2} - 3 x - 2 - 3 x^{3} + 5 x^{2} 0 - 9 x^{2} + 9 x + 10 9 x^{2} - 15 x 0 - 6 x + 10 6 x - 10 0 \Rightarrow 3 x^{3} - 14 x^{2} + 9 x + 10 = (3 x - 5) (x^{2} - 3 x - 2) C o n s i d e r (x^{2} - 3 x - 2) x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{3 \pm \sqrt{9 + 8}}{2} = \frac{3 \pm \sqrt{17}}{2} \Rightarrow (2 x - 3 + \sqrt{17}) a n d (2 x - 3 - \sqrt{17}) a r e f a c t o r s o f (x^{2} - 3 x - 2) \Rightarrow 3 x^{3} - 14 x^{2} + 9 x + 10 = (3 x - 5) (2 x - 3 + \sqrt{17}) (2 x - 3 - \sqrt{17})$

4)
$15 x^{3} - 34 x^{2} + 21 x - 10 U \sin g t r i a l a n d e r r o r m e t h o d w e g e t t h a t 15 x^{3} - 34 x^{2} + 21 x - 10 = 0 w h e n x = \frac{5}{3} \Rightarrow (3 x - 5) i s a f a c t o r o f 15 x^{3} - 34 x^{2} + 21 x - 10 U \sin g l o n g d i v i s i o n w e g e t \sqrt[3 x - 5]{15 x^{3} - 34 x^{2} + 21 x - 10} 5 x^{2} - 3 x + 2 - 15 x^{3} + 25 x^{2} 0 - 9 x^{2} + 21 x - 10 9 x^{2} - 15 x 0 + 6 x - 10 - 6 x + 10 0 \Rightarrow 15 x^{3} - 34 x^{2} + 21 x - 10 = (3 x - 5) (5 x^{2} - 3 x + 2)$

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