Math Problem Statement

Divide: (a^2 + 14a + 45) / (a + 9)

Solution

The given expression is:

a2+14a+45a+9\frac{a^2 + 14a + 45}{a + 9}

To divide this expression, we need to factor the numerator and simplify if possible.

Step 1: Factor the numerator a2+14a+45a^2 + 14a + 45.

We are looking for two numbers that multiply to 4545 (the constant term) and add to 1414 (the coefficient of the linear term). The numbers that satisfy this are 99 and 55, because:

9×5=45and9+5=149 \times 5 = 45 \quad \text{and} \quad 9 + 5 = 14

Thus, the factored form of the numerator is:

a2+14a+45=(a+9)(a+5)a^2 + 14a + 45 = (a + 9)(a + 5)

Step 2: Simplify the expression.

Now substitute the factored form into the original expression:

(a+9)(a+5)a+9\frac{(a + 9)(a + 5)}{a + 9}

Cancel the common factor a+9a + 9 from both the numerator and denominator (assuming a9a \neq -9 to avoid division by zero):

(a+9)(a+5)a+9=a+5\frac{(a + 9)(a + 5)}{a + 9} = a + 5

Final Answer:

The simplified expression is:

a+5a + 5

Would you like further details or have any questions?

Here are 5 related questions:

  1. How do you factor quadratic expressions?
  2. What happens if there is no common factor in such division problems?
  3. How can you verify if your factored form is correct?
  4. What is the condition for cancelling terms in rational expressions?
  5. Can you use synthetic division for such problems? When would that be appropriate?

Tip: Always check for restrictions in the variable when cancelling terms in rational expressions (like ensuring the denominator is not zero).

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Division
Factoring Quadratics

Formulas

Factoring a quadratic expression: (a^2 + bx + c) = (a + p)(a + q)

Theorems

Factoring Theorem
Cancellation in Rational Expressions

Suitable Grade Level

Grades 8-10