What is the factored form of the polynomial? x2 – 12x + 27? O (x + 4)(x + 3) (x – 4)(x + 3) O (X + 9)(x +ht (x – 9)(x-3) o
The Correct Answer and Explanation is:
To factor the quadratic polynomial x2−12x+27x^2 – 12x + 27×2−12x+27, we need to express it as the product of two binomials of the form (x−a)(x−b)(x – a)(x – b)(x−a)(x−b). Here’s how we can factor it step-by-step:
Step 1: Look for two numbers that multiply to give the constant term (27) and add to give the coefficient of the middle term (-12).
The two numbers should satisfy the following conditions:
- Their product must be equal to the constant term, 27.
- Their sum must be equal to the middle term’s coefficient, -12.
Step 2: Find factor pairs of 27.
The factor pairs of 27 are:(1,27),(3,9),(−1,−27),(−3,−9)(1, 27), (3, 9), (-1, -27), (-3, -9)(1,27),(3,9),(−1,−27),(−3,−9)
Step 3: Check which pair sums to -12.
From the factor pairs above, the pair that adds up to -12 is (−3)(-3)(−3) and (−9)(-9)(−9), since:−3+(−9)=−12-3 + (-9) = -12−3+(−9)=−12
And also:−3×−9=27-3 \times -9 = 27−3×−9=27
Step 4: Write the factored form.
Thus, the quadratic expression can be factored as:(x−3)(x−9)(x – 3)(x – 9)(x−3)(x−9)
Conclusion:
The correct factorization of x2−12x+27x^2 – 12x + 27×2−12x+27 is (x−3)(x−9)(x – 3)(x – 9)(x−3)(x−9).
Correct answer: (x−9)(x−3)(x – 9)(x – 3)(x−9)(x−3).
This is the factored form, and the reasoning involves finding two numbers that multiply to 27 and add up to -12, allowing us to factor the quadratic expression effectively.
