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(at+t^(2))/(2b-ct)÷(a^(2)+2at+t^(2))/(2bt-ct^(2))

$5$ ) $\frac{a t+t^{2}}{2 b-c t} \div \frac{a^{2}+2 a t+t^{2}}{2 b t-c t^{2}}$

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Divide polynomials using long division

Full solution

Q.

5

)

\frac{a t+t^{2}}{2 b-c t} \div \frac{a^{2}+2 a t+t^{2}}{2 b t-c t^{2}}

Rewrite as reciprocal multiplication: Rewrite the division as multiplication by the reciprocal of the second fraction. $\newline$ $(at + t^2) / (2b - ct) \times (2bt - ct^2) / (a^2 + 2at + t^2)$
Factor numerator and denominator: Factor the numerator and denominator of the second fraction. $(at + t^2) / (2b - ct) \times (2bt - ct^2) / ((a + t)^2)$
Identify common factors: Notice that $(at + t^2)$ is the same as $t(a + t)$ and $(a + t)^2$ is $(a + t)(a + t)$ . $\frac{t(a + t)}{(2b - ct)} \cdot \frac{(2bt - ct^2)}{(a + t)(a + t)}$
Cancel common factor: Cancel out the common factor of $(a + t)$ in the numerator of the first fraction and the denominator of the second fraction. $\frac{t}{(2b - ct)} \times \frac{(2bt - ct^2)}{(a + t)}$
Expand remaining terms: Expand the remaining terms in the second fraction's numerator. $\frac{t}{2b - ct} \cdot \frac{2bt - c t^2}{a + t}$
Multiply numerators and denominators: Now multiply the numerators and denominators. $\frac{t \times (2bt - c \times t^2)}{(2b - ct) \times (a + t)}$
Distribute $t$ in numerator: Distribute $t$ in the numerator. $\newline$ $(2b\cdot t^2 - c\cdot t^3) / ((2b - ct) \cdot (a + t))$
Distribute denominator: Distribute the denominator. $\newline$ $(2b*t^2 - c*t^3) / (2ab + 2bt - act - c*t^2)$
Combine like terms: Combine like terms in the denominator. $\newline$ $(2b*t^2 - c*t^3) / (2ab + 2bt - act - c*t^2)$ $\newline$ Oops, there's a mistake here. The term $-act$ should be $-cat$ .

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