Question

unit 4a - polynomial functions
simplify. all answers should be in standard form and contain only positive exponents.

1. (y + 3)(2y² - y - 4)
2. (12x⁵y³ - 20x⁸y² + 4x⁷y) / (4x²y)

Explanation:

Step1: Distribute $y$ into trinomial

$y\cdot2y^2 + y\cdot(-y) + y\cdot(-4) = 2y^3 - y^2 -4y$

Step2: Distribute $3$ into trinomial

$3\cdot2y^2 + 3\cdot(-y) + 3\cdot(-4) = 6y^2 -3y -12$

Step3: Combine terms and simplify

$2y^3 - y^2 -4y +6y^2 -3y -12 = 2y^3 +5y^2 -7y -12$

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Step1: Split fraction into three terms

$\frac{12x^5y^3}{4x^2y} - \frac{20x^8y^2}{4x^2y} + \frac{4x^2y}{4x^2y}$

Step2: Simplify each term

$3x^{5-2}y^{3-1} -5x^{8-2}y^{2-1} +1 = 3x^3y^2 -5x^6y +1$

Step3: Rearrange to standard form

$-5x^6y +3x^3y^2 +1$

Answer:

1. $2y^3 +5y^2 -7y -12$
2. $-5x^6y +3x^3y^2 +1$