Breaking Down (c² + 2c - 35) ÷ (c - 5) Step by Step

Let's perform the polynomial long division step by step.First, we divide the leading term c squared by c to get c.Now multiply c times c minus 5 to get c squared minus 5c.Subtracting gives us 7c minus 35.Next, we divide 7c by c to get 7.Multiply 7 times c minus 5 to get 7c minus 35.When we subtract, all terms cancel out perfectly, giving us zero. This means there is no remainder.Our polynomial division gave us c plus 7 as the simplified answer.To verify this is correct, let's multiply our answer by the denominator. This should give us back the numerator.We multiply c plus 7 times c minus 5 using the FOIL method.First, multiply each term: c times c, c times negative 5, 7 times c, and 7 times negative 5.This gives us c squared, negative 5c, plus 7c, minus 35.Combining like terms, negative 5c plus 7c equals positive 2c.This simplifies to c squared plus 2c minus 35, which matches our original numerator exactly!However, we need to be careful about the domain of our expression.The original expression is undefined when c equals 5, because we can't divide by zero.Our simplified form, c plus 7, must also exclude c equals 5 to maintain equivalence.For all other values of c, both expressions are completely equivalent.Our verification proves that c plus 7 is indeed the correct simplified form of our rational expression.