Curious about the factored form of the polynomial x^2 – 12x + 27? Let me break it down for you. The factored form represents a polynomial as a product of its factors, which are expressions that can be multiplied together to obtain the original polynomial. In this case, we’ll look at how to factorize x^2 – 12x + 27.
To find the factored form, we need to identify two binomials that, when multiplied together, give us the original polynomial. We can start by looking for two numbers whose product is equal to the constant term (in this case, 27) and whose sum is equal to the coefficient of the middle term (which is -12 in our polynomial).
What is the Factored Form of the Polynomial? x^2 – 12x + 27?
Factored Form: Definition and Explanation
The factored form of a polynomial represents it as a product of its factors. In this case, we are looking at the polynomial x^2 – 12x + 27. The factored form allows us to express the polynomial in a way that makes it easier to analyze and solve.
To understand how to convert a polynomial into factored form, let’s first break down the components of this specific equation. The given polynomial has three terms: x^2, -12x, and 27. Our goal is to rewrite it in factored form using these terms.
Steps to Convert a Polynomial into Factored Form
- Identify any common factors: Look for any common factors among all three terms. In our example, there are no common factors other than “1.”
- Factor the quadratic term: Focus on the quadratic term (x^2) and try to find two binomial factors that multiply together to give x^2. Since there is no coefficient attached to x^2, we can directly factor it as (x)(x).
- Factor the remaining terms: Now we need to factorize -12x and 27 separately. Let’s examine -12x first. We look for two numbers whose product gives us -12 (-3 * 4) and whose sum or difference matches the coefficient of x (-3 + 4 = -12). So, we can rewrite -12x as (-3x)(4).
- Combine all factors: Finally, combine all the factored terms obtained in steps two and three:
(x)(-3x)(4) = -12(x^2)
Therefore, rewriting our equation with these steps yields:
x^2 – 12x + 27 = (x)(-3x)(4) + 27
Exploring Polynomial Equations
Polynomial equations are an essential part of algebraic mathematics, and understanding them is crucial in various fields such as physics, engineering, and computer science. In this section, I’ll provide insights into the concept of factored form for polynomial equations and specifically address the question: “What is the factored form of the polynomial x^2 – 12x + 27?”
To determine the factored form of a polynomial equation like this one, we need to factorize it into its simplest possible terms. The factored form represents an expression written as a product of its factors. In this case, we aim to break down the quadratic equation x^2 – 12x + 27 into its factors.
Firstly, let’s analyze the given equation by identifying its coefficients:
Coefficient of x^2: 1 Coefficient of x: -12 Constant term: 27
When factoring quadratic equations like this one, we look for two binomials that multiply together to give us our original equation. In other words, we want to find two expressions whose product equals x^2 – 12x + 27.
By examining the coefficients and constant term, we can deduce that our desired factors must satisfy certain conditions:
- Their product should be equal to the constant term (27).
- Their sum should be equal to the coefficient of x (-12).
Considering these conditions and utilizing techniques such as trial and error or quadratic formula calculations, we can determine that our factored form for x^2 – 12x + 27 is:
(x – 3)(x – 9)
This means that (x – 3) and (x -9) are our factors which when multiplied together yield our original polynomial equation.
To verify whether this is indeed correct, we can expand (multiply) out these factors using FOIL (First Outer Inner Last) method:
(x – 3)(x – 9) = x^2 – 9x – 3x + 27 = x^2 – 12x + 27
As you can see, the expanded form matches our original equation, confirming that (x – 3)(x -9) is indeed the factored form of x^2 – 12x + 27.
Understanding how to factorize polynomial equations is essential as it allows us to simplify complex expressions and solve equations more efficiently. Factoring also helps in identifying roots or zeros of the equation, which are crucial in various mathematical applications.
