What is the factored form of the polynomial?
x2 − 12x + 27?
The Correct Answer and Explanation is :
To factor the polynomial (x^2 – 12x + 27), we can use the method of finding two numbers that multiply to the constant term (27) and add up to the coefficient of the linear term (-12).
- Identify the Terms: The polynomial is in the standard quadratic form (ax^2 + bx + c), where:
- (a = 1)
- (b = -12)
- (c = 27)
- Find the Product and Sum: We need to find two numbers that:
- Multiply to (c) (27)
- Add to (b) (-12)
- Listing Factor Pairs of 27: The factor pairs of 27 are:
- (1 \times 27)
- (3 \times 9)
- Evaluate Factor Pairs: Next, we check which pair adds up to -12:
- (1 + 27 = 28)
- (3 + 9 = 12) We notice that neither pair yields -12. However, since we need the factors to also be negative (as the linear term’s coefficient is negative), we can instead consider:
- (-3) and (-9)
- Confirm the Factors:
- Multiply: ((-3) \times (-9) = 27)
- Add: ((-3) + (-9) = -12) These values meet both criteria.
- Write the Factored Form: We can now express the polynomial in its factored form. The original polynomial (x^2 – 12x + 27) can be factored as:
[
(x – 3)(x – 9)
] - Verification: To ensure our factorization is correct, we can expand ( (x – 3)(x – 9) ):
- Using the distributive property:
[
(x – 3)(x – 9) = x^2 – 9x – 3x + 27 = x^2 – 12x + 27
]
This confirms our factored form is correct.
In conclusion, the polynomial (x^2 – 12x + 27) can be factored into the form ((x – 3)(x – 9)). This process highlights the importance of finding appropriate factors that satisfy both multiplication and addition requirements, leading to an accurate and simplified expression.