What Is The Factored Form Of The Polynomial X²-16X+48?

What is the factored form of the polynomial? X^216x+48 from brainly.com

Understanding Polynomials

Before we dive into the factored form of the polynomial x²-16x+48, let's first understand what a polynomial is. Simply put, a polynomial is an expression consisting of variables and coefficients, which are combined using addition, subtraction, multiplication, and division. These expressions are often used in algebra, and they can have various forms, such as quadratic, cubic, and quartic.

The Quadratic Polynomial

The quadratic polynomial is a type of polynomial that has a degree of 2, meaning it has a maximum of two terms that contain the variable. The general form of a quadratic polynomial is ax²+bx+c, where a, b, and c are constants. The term "quadratic" comes from the Latin word "quadratus," which means "square," as the variable is squared in this type of polynomial.

Factoring a Quadratic Polynomial

Now that we understand what a polynomial is, let's focus on factoring a quadratic polynomial. Factoring is the process of breaking down a polynomial into its factors, which are expressions that can be multiplied together to yield the original polynomial. Factoring a quadratic polynomial is particularly useful as it allows us to solve equations that involve the given polynomial.

The Factored Form of a Quadratic Polynomial

The factored form of a quadratic polynomial is an expression that shows the polynomial as a product of its factors. For example, the factored form of x²-4x+3 is (x-3)(x-1). In this form, we can easily see the factors of the polynomial, which are (x-3) and (x-1).

Factoring x²-16x+48

Now, let's focus on factoring the given polynomial, x²-16x+48. To factor this polynomial, we need to find two factors that multiply to give us 48 and add up to -16. One way to do this is to list all the factors of 48 and check which two add up to -16. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. After trying different combinations, we can see that 4 and 12 add up to -16 and multiply to give us 48. Therefore, we can write the factored form of x²-16x+48 as (x-4)(x-12).

Checking the Factored Form

To check if the factored form of a polynomial is correct, we can use the distributive property of multiplication. We can expand the factored form and see if it yields the original polynomial. Let's do this for x²-16x+48. (x-4)(x-12) = x(x-12)-4(x-12) = x²-12x-4x+48 = x²-16x+48 As we can see, the expanded form of the factored expression is equal to the original polynomial. Therefore, we can conclude that the factored form of x²-16x+48 is (x-4)(x-12).

Using the Factored Form

Now that we have the factored form of x²-16x+48, we can use it to solve equations that involve this polynomial. For example, if we are asked to find the values of x that make the polynomial equal to zero, we can set the factored expression equal to zero and solve for x. (x-4)(x-12) = 0 x-4 = 0 or x-12 = 0 x = 4 or x = 12 Therefore, the values of x that make x²-16x+48 equal to zero are 4 and 12.

Conclusion

In conclusion, the factored form of the polynomial x²-16x+48 is (x-4)(x-12). Factoring a quadratic polynomial involves finding two factors that multiply to give us the constant term and add up to give us the coefficient of the middle term. The factored form of a polynomial can be used to solve equations that involve the polynomial.