Simplify the expression completely if possible. (5x^(4)+15x^(3))/(x^(2)-6x-27) Answer:

Factor Numerator: First, we will try to factor both the numerator and the denominator to see if any terms can be canceled out. Starting with the numerator: 5x^(4)+15x^(3) can be factored by taking out the common factor of 5x^3. 5x^(4)+15x^(3) = 5x^3(x+3)Factor Denominator: Now, let's factor the denominator: x^(2)-6x-27 is a quadratic expression, and we will look for two numbers that multiply to -27 and add up to -6. The numbers -9 and 3 satisfy these conditions. So we can write the denominator as: x^(2)-6x-27 = (x-9)(x+3)Cancel Common Term: Now we have the expression in a factored form: (5x^3(x+3))/((x-9)(x+3)) We can see that the (x+3) term is present in both the numerator and the denominator, so we can cancel it out.Final Simplification: After canceling out the (x+3) term, we are left with: (5x^3)/(x-9) This is the simplified form of the original expression, as no further simplification is possible.

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Q. Simplify the expression completely if possible.

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\frac{5 x^{4}+15 x^{3}}{x^{2}-6 x-27}

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Answer:

Factor Numerator: First, we will try to factor both the numerator and the denominator to see if any terms can be canceled out. $\newline$ Starting with the numerator: $\newline$ $5x^{4}+15x^{3}$ can be factored by taking out the common factor of $5x^3$ . $\newline$ $5x^{4}+15x^{3} = 5x^3(x+3)$
Factor Denominator: Now, let's factor the denominator: $\newline$ $x^2-6x-27$ is a quadratic expression, and we will look for two numbers that multiply to $-27$ and add up to $-6$ . $\newline$ The numbers $-9$ and $3$ satisfy these conditions. $\newline$ So we can write the denominator as: $\newline$ $x^2-6x-27 = (x-9)(x+3)$
Cancel Common Term: Now we have the expression in a factored form: $\newline$ $(5x^3(x+3))/((x-9)(x+3))$ $\newline$ We can see that the $(x+3)$ term is present in both the numerator and the denominator, so we can cancel it out.
Final Simplification: After canceling out the $(x+3)$ term, we are left with: $\frac{5x^3}{x-9}$ This is the simplified form of the original expression, as no further simplification is possible.