Factor the trinomial completely. $$ p^{2}-12 p+35 $$

Question

UpStudy AI · Accepted Answer

To factor the trinomial $$ p^{2} - 12p + 35 $$ completely, follow these steps:

### Step 1: Identify the Coefficients
The given quadratic trinomial is in the form:
$$ p^{2} + bp + c $$
where:
- $$ b = -12 $$
- $$ c = 35 $$

### Step 2: Find Two Numbers That Multiply to $$ c $$ and Add to $$ b $$
We need to find two numbers, let's call them $$ m $$ and $$ n $$, such that:
$$ m \times n = c = 35 $$
$$ m + n = b = -12 $$

Let's list the pairs of factors of 35:
- $$ 1 $$ and $$ 35 $$
- $$ 5 $$ and $$ 7 $$

Since the constant term $$ c $$ is positive and the middle term $$ b $$ is negative, both numbers $$ m $$ and $$ n $$ must be negative.

Check the pair $$ -5 $$ and $$ -7 $$:
- $$ -5 \times -7 = 35 $$ ✅
- $$ -5 + (-7) = -12 $$ ✅

### Step 3: Write the Factored Form
Using the numbers $$ -5 $$ and $$ -7 $$, the trinomial can be factored as:
$$ p^{2} - 12p + 35 = (p - 5)(p - 7) $$

### Verification
To ensure the factorization is correct, expand the factored form:
$$
(p - 5)(p - 7) = p \times p + p \times (-7) + (-5) \times p + (-5) \times (-7) \
= p^{2} - 7p - 5p + 35 \
= p^{2} - 12p + 35
$$
The expanded form matches the original trinomial, confirming that the factorization is correct.

### Final Answer
$$ p^{2} - 12p + 35 = (p - 5)(p - 7) $$
$$ p^{2} - 12p + 35 = (p - 5)(p - 7) $$

Factor the trinomial completely.

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Solución

Step 1: Identify the Coefficients

Step 2: Find Two Numbers That Multiply to and Add to

Step 3: Write the Factored Form

Verification

Final Answer

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