Problem 4 - Elementary Algebra

Question

Reduce each of the following fractions to lowest terms. $$ \frac{-3 a^{4}+75 a^{2}}{2 a^{3}-16 a^{2}+30 a} $$

Step-by-Step Solution

Answer

Question: Reduce the following fraction to its lowest terms: $$ \frac{-3 a^{4}+75 a^{2}}{2 a^{3}-16 a^{2}+30 a}. $$ Answer: The reduced fraction is: $$ \frac{3(a + 5)}{2(a - 3)}. $$

Step 1: Factor out common terms from numerator and denominator

In the given fraction, $$ \frac{-3 a^{4}+75 a^{2}}{2 a^{3}-16 a^{2}+30 a}, $$ we observe that the numerator has a common factor of \(3a^2\) and the denominator has a common factor of \(2a\). Let's factor these out: $$ \frac{3a^2(-a^{2}+25)}{2a(a^{2}-8a+15)}. $$

Step 2: Factorize the numerator and the denominator further

We can factorize the numerator further by using the difference of squares formula: $$ (-a^2 + 25) = (a + 5)(-a + 5). $$ For the denominator, we can try to factorize the quadratic expression \(a^2-8a+15\): $$ (a^2 - 8a + 15) = (a - 3)(a - 5). $$ So now, our fraction looks like: $$ \frac{3a^2(a + 5)(-a + 5)}{2a(a-3)(a-5)}. $$

Step 3: Cancel out any common factors

Let's cancel out the common factors from the numerator and the denominator of the fraction: $$ \require{cancel} \frac{3\cancel{a^2}(a + 5)(\cancel{-a + 5})}{2\cancel{a}(a-3)(\cancel{a-5})}. $$ After cancelling common factors, we are left with the reduced fraction: $$ \frac{3(a + 5)}{2(a - 3)}. $$