Q. Fully simplify the expression below and write your answer as a single fraction.9x2+81x+723x+3⋅x2−2x−48x2−64Answer:
Factor Common Terms: First, factor each polynomial in the expression to see if any terms can be canceled out.Starting with the numerator of the first fraction, factor out the common factor of 3.(3x+3)=3(x+1)
Factor Numerator 1: Now, factor the denominator of the first fraction. The quadratic 9x2+81x+72 can be factored into two binomials.9x2+81x+72=(3x+9)(3x+8)
Factor Denominator 1: Next, factor the numerator of the second fraction. The difference of squares x2−64 can be factored into (x+8)(x−8). x2−64=(x+8)(x−8)
Factor Numerator 2: Now, factor the denominator of the second fraction. The quadratic x2−2x−48 can be factored into two binomials.x2−2x−48=(x−8)(x+6)
Factor Denominator 2: Now that we have factored all parts of the expression, we can write the entire expression with these factors and look for common terms to cancel out.3(x+9)(x+8)3(x+1)×(x−8)(x+6)(x+8)(x−8)
Combine Factors: We can cancel out the common terms (x+8) and (x−8) from the numerator and denominator.The expression simplifies to:3(x+9)(x+6)3(x+1)
Cancel Common Terms: We can also cancel out the common factor of 3 from the numerator and denominator.The expression further simplifies to:(x+9)(x+6)x+1
Further Simplify: Now we have the fully simplified expression as a single fraction.The final answer is (x+1)/((x+9)(x+6)).
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