Factortsed form of `r ^(2) - 10 r + 21 ` is

Options

A. `(r-1) ( r -4)`
B. `(r - 7) (r -3)`
C. `(r -7) ( r + 3)`
D. `(r + 7) ( r + 3)`

Answer

To factor the expression \( r^2 - 10r + 21 \), we will follow these steps:

### Step 1: Identify the coefficients
In the expression \( r^2 - 10r + 21 \), we identify:
- The coefficient of \( r^2 \) (which is 1),
- The coefficient of \( r \) (which is -10),
- The constant term (which is 21).

### Step 2: Find two numbers that multiply and add to specific values
We need to find two numbers \( a \) and \( b \) such that:
- The sum \( a + b = -10 \) (the coefficient of \( r \)),
- The product \( a \cdot b = 21 \) (the constant term).

### Step 3: List pairs of factors of 21
The pairs of factors of 21 are:
- \( 1 \times 21 \)
- \( 3 \times 7 \)

### Step 4: Determine the correct pair
We need to check which pair can add up to -10 when both numbers are negative:
- \( -3 \) and \( -7 \) (since \( -3 + -7 = -10 \) and \( -3 \cdot -7 = 21 \)).

### Step 5: Rewrite the expression
Now we can rewrite the middle term using \( -3 \) and \( -7 \):
\[ r^2 - 3r - 7r + 21 \]

### Step 6: Group the terms
Next, we group the terms:
\[ (r^2 - 3r) + (-7r + 21) \]

### Step 7: Factor out the common terms
Now, we factor out the common terms from each group:
\[ r(r - 3) - 7(r - 3) \]

### Step 8: Factor by grouping
Now we can factor out the common binomial factor \( (r - 3) \):
\[ (r - 3)(r - 7) \]

### Final Answer
Thus, the factored form of \( r^2 - 10r + 21 \) is:
\[ (r - 3)(r - 7) \]

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