What is the factored form of the polynomial? x2 – 12x + 27
The Correct Answer and Explanation is:
The factored form of the quadratic polynomial x2−12x+27x^2 – 12x + 27×2−12x+27 is:(x−3)(x−9)(x – 3)(x – 9)(x−3)(x−9)
Explanation:
To factor the quadratic expression x2−12x+27x^2 – 12x + 27×2−12x+27, we need to find two numbers that:
- Multiply to give the constant term (27).
- Add up to give the coefficient of the linear term (-12).
Let’s break it down step by step:
- Start with the general form of a quadratic equation: ax2+bx+cax^2 + bx + cax2+bx+c In our case, a=1a = 1a=1, b=−12b = -12b=−12, and c=27c = 27c=27.
- Look for two numbers that multiply to ac=1×27=27ac = 1 \times 27 = 27ac=1×27=27 and add up to b=−12b = -12b=−12. We need two numbers that multiply to 27 and add to -12. These numbers are -3 and -9 because: (−3)×(−9)=27(-3) \times (-9) = 27(−3)×(−9)=27 (−3)+(−9)=−12(-3) + (-9) = -12(−3)+(−9)=−12
- Rewrite the middle term using these two numbers:
The middle term, −12x-12x−12x, can be rewritten as −3x−9x-3x – 9x−3x−9x. Now, we have: x2−3x−9x+27x^2 – 3x – 9x + 27×2−3x−9x+27 - Factor by grouping:
Group the terms in pairs: (x2−3x)−(9x−27)(x^2 – 3x) – (9x – 27)(x2−3x)−(9x−27) Factor out the greatest common factor (GCF) from each pair: x(x−3)−9(x−3)x(x – 3) – 9(x – 3)x(x−3)−9(x−3) - Factor out the common binomial:
Now that both terms contain the factor (x−3)(x – 3)(x−3), factor it out: (x−3)(x−9)(x – 3)(x – 9)(x−3)(x−9)
Thus, the factored form of x2−12x+27x^2 – 12x + 27×2−12x+27 is (x−3)(x−9)(x – 3)(x – 9)(x−3)(x−9).
