Question

describe the end behavior of the graph of the following polynomial function:
\\( p(x)=2x^4 + x^3 + 6x^2 - 10 \\)
\\( \bigcirc \\) falls to the left and rises to the right
\\( \bigcirc \\) rises to the left and rises to the right
\\( \bigcirc \\) falls to the left and falls to the right
\\( \bigcirc \\) rises to the left and falls to the right

question 33
simplify completely:
\\( \frac{18m^2 + 54m + 36}{9m^2 + 45m + 54} \\)
\\( \bigcirc \\) \\( \frac{2(m^2 + 3m + 2)}{m^2 + 5m + 6} \\)
\\( \bigcirc \\) \\( \frac{2(m + 1)}{m + 3} \\)
\\( \bigcirc \\) \\( \frac{m + 1}{m + 3} \\)
\\( \bigcirc \\) \\( \frac{18(m + 1)}{m + 3} \\)

Explanation:

First Problem: End Behavior of Polynomial

Step1: Identify degree and leading coefficient

Degree = 4 (even), leading coefficient = 2 (positive)

Step2: Apply end behavior rules

For even degree, positive leading coefficient: as $x\to-\infty$, $p(x)\to+\infty$; as $x\to+\infty$, $p(x)\to+\infty$

Step1: Factor numerator and denominator

Numerator: $18m^2+54m+36 = 18(m^2+3m+2) = 18(m+1)(m+2)$
Denominator: $9m^2+45m+54 = 9(m^2+5m+6) = 9(m+2)(m+3)$

Step2: Cancel common factors

Cancel $9(m+2)$ from numerator/denominator:
$\frac{18(m+1)(m+2)}{9(m+2)(m+3)} = \frac{2(m+1)}{m+3}$

Answer:

Rises to the left and rises to the right

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Second Problem: Simplify Rational Expression