Factor The Trinomial: C 2 − 12 C + 35 C^2 - 12c + 35 C 2 − 12 C + 35 Select The Correct Choice Below And, If Necessary, Fill In The Answer Box Within Your Choice.A. The Answer Is { \square$}$. (Factor Completely.) B. The Trinomial Is Not Factorable.

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**Factor the Trinomial: $c^2 - 12c + 35$**

Understanding the Problem

Factoring a trinomial is a process of expressing it as a product of two binomials. In this case, we are given the trinomial c2−12c+35c^2 - 12c + 35 and we need to factor it completely.

Step 1: Identify the Coefficients

To factor the trinomial, we need to identify the coefficients of the three terms. The coefficients are the numbers that multiply the variables. In this case, the coefficients are:

  • c2c^2 has a coefficient of 1
  • −12c-12c has a coefficient of -12
  • +35+35 has a coefficient of 35

Step 2: Look for Two Numbers Whose Product is 35 and Whose Sum is -12

To factor the trinomial, we need to find two numbers whose product is 35 and whose sum is -12. These numbers are -7 and -5, because:

  • −7×−5=35-7 \times -5 = 35
  • −7+(−5)=−12-7 + (-5) = -12

Step 3: Write the Trinomial as a Product of Two Binomials

Now that we have found the two numbers, we can write the trinomial as a product of two binomials:

c2−12c+35=(c−7)(c−5)c^2 - 12c + 35 = (c - 7)(c - 5)

Answer

The correct answer is:

A. The answer is (c−7)(c−5)\boxed{(c - 7)(c - 5)}.

Discussion

Factoring a trinomial is an important skill in algebra, and it can be used to solve equations and inequalities. In this case, we used the method of factoring by grouping to factor the trinomial c2−12c+35c^2 - 12c + 35. This method involves looking for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q&A

Q: What is the first step in factoring a trinomial? A: The first step in factoring a trinomial is to identify the coefficients of the three terms.

Q: How do we find the two numbers whose product is 35 and whose sum is -12? A: We can find the two numbers by trial and error, or by using the method of factoring by grouping.

Q: What is the final answer to the problem? A: The final answer is (c−7)(c−5)\boxed{(c - 7)(c - 5)}.

Q: Can a trinomial always be factored? A: No, a trinomial is not always factorable. If the trinomial cannot be factored, it is said to be irreducible.

Q: What is the importance of factoring a trinomial? A: Factoring a trinomial is an important skill in algebra, and it can be used to solve equations and inequalities.

Q: How do we use factoring to solve equations and inequalities? A: We can use factoring to solve equations and inequalities by setting the factored expression equal to zero and solving for the variable.

Q: What are some common mistakes to avoid when factoring a trinomial? A: Some common mistakes to avoid when factoring a trinomial include:

  • Not identifying the coefficients of the three terms
  • Not finding the two numbers whose product is the constant term and whose sum is the coefficient of the linear term
  • Not writing the trinomial as a product of two binomials

Q: How do we check our answer when factoring a trinomial? A: We can check our answer by multiplying the two binomials together and making sure that the result is equal to the original trinomial.