Question
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c) \( \left(2 x^{3} y^{2}-3 x^{2} y^{4}+x y\right)(5 x-2) \)

Ask by Ellis Gibson. in Mexico
Nov 12,2024

Real Tutor Solution

Tutor-Verified Answer

Answer

La expresión simplificada es \(10x^{4}y^{2}-4x^{3}y^{2}-15x^{3}y^{4}+6x^{2}y^{4}+5x^{2}y-2xy\).

Solution

Calculate or simplify the expression \( (2*x^3*y^2-3*x^2*y^4+x*y)*(5*x-2) \). Simplify the expression by following steps: - step0: Solution: \(\left(2x^{3}y^{2}-3x^{2}y^{4}+xy\right)\left(5x-2\right)\) - step1: Apply the distributive property: \(2x^{3}y^{2}\times 5x-2x^{3}y^{2}\times 2-3x^{2}y^{4}\times 5x-\left(-3x^{2}y^{4}\times 2\right)+xy\times 5x-xy\times 2\) - step2: Multiply the terms: \(10x^{4}y^{2}-4x^{3}y^{2}-15x^{3}y^{4}-\left(-6x^{2}y^{4}\right)+5x^{2}y-2xy\) - step3: Remove the parentheses: \(10x^{4}y^{2}-4x^{3}y^{2}-15x^{3}y^{4}+6x^{2}y^{4}+5x^{2}y-2xy\) La expresión simplificada es \(10x^{4}y^{2}-4x^{3}y^{2}-15x^{3}y^{4}+6x^{2}y^{4}+5x^{2}y-2xy\).

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The Deep Dive

To multiply the expression \( \left(2 x^{3} y^{2}-3 x^{2} y^{4}+x y\right)(5 x-2) \), we can use the distributive property, also known as the FOIL method (First, Outside, Inside, Last) for polynomials. Here's how it goes: First, distribute \(5x\) over each term in the polynomial: 1. \(5x \cdot 2x^3y^2 = 10x^4y^2\) 2. \(5x \cdot (-3x^2y^4) = -15x^3y^4\) 3. \(5x \cdot xy = 5x^2y\) Next, distribute \(-2\) over each term in the polynomial: 1. \(-2 \cdot 2x^3y^2 = -4x^3y^2\) 2. \(-2 \cdot (-3x^2y^4) = 6x^2y^4\) 3. \(-2 \cdot xy = -2xy\) Now, combine all the terms together: \[ 10x^4y^2 + (-15x^3y^4) + 5x^2y + (-4x^3y^2) + 6x^2y^4 - 2xy \] Now, we group like terms: - The \(x^4y^2\) term: \(10x^4y^2\) - The \(x^3y^4\) terms: \(-15x^3y^4 - 4x^3y^2 = -19x^3y^2\) - The \(x^2y\) term: \(5x^2y + 6x^2y^4 = 11x^2y^4\) - The \(xy\) term: \(-2xy\) Thus, the final expression simplifies to: \[ 10x^4y^2 - 19x^3y^2 + 11x^2y^4 - 2xy \]

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