Multiply the factors found above to find the simplified polynomial function.
[tex]f(x)=(x+2)(+2)(x+3)(3)[/tex] Write the function as a product of its factors.
[tex]
\begin{array}{c}
f(x)=(x+2)(+3)(x+3)(3) \\
f(x)=(x+3)(x+3)(3+6) \\
\downarrow \\
\downarrow \\
\left.f(x)=(F) x=3 x^2\right)(\quad)
\end{array}
[/tex]
Use the Distributive Property to multiply the first two real factors. Do not forget to simplify?
Use the Distributive Property to multiply the last two complex factors. Do not forget to simplify!
Use the Distributive Property a third time to
[tex]P(x)=[/tex]
multiply the real and the complex parts. Make sure to simplify. Remember that [tex]i^2=-1[/tex].
Answer (2)
To simplify the function f ( x ) = ( x + 2 ) ( 2 ) ( x + 3 ) ( 3 ) , we first multiply the constant terms to get 6. Then, we multiply the binomials to find ( x + 2 ) ( x + 3 ) = x 2 + 5 x + 6 and multiply the result by 6, yielding the final polynomial f ( x ) = 6 x 2 + 30 x + 36 .
;
Multiply the constant terms: 2 × 3 = 6 .
Multiply the binomial factors: ( x + 2 ) ( x + 3 ) = x 2 + 5 x + 6 .
Multiply the result by the constant: 6 ( x 2 + 5 x + 6 ) = 6 x 2 + 30 x + 36 .
The simplified polynomial function is: 6 x 2 + 30 x + 36 .
Explanation
Analyzing the Expression Let's analyze the given expression and the instructions. The expression is presented as f ( x ) = ( x + 2 ) ( + 2 ) ( x + 3 ) ( 3 ) . There appear to be some formatting issues, particularly with the (+2) and (+3) terms. Assuming these are meant to be constants, we can interpret the function as f ( x ) = ( x + 2 ) ⋅ 2 ⋅ ( x + 3 ) ⋅ 3 . The goal is to simplify this expression into a polynomial function P ( x ) using the distributive property.
Multiplying Constants First, let's multiply the constant terms together: 2 × 3 = 6 . So, we can rewrite the function as f ( x ) = 6 ( x + 2 ) ( x + 3 ) .
Multiplying Binomials Next, we'll multiply the binomial factors ( x + 2 ) and ( x + 3 ) using the distributive property (also known as the FOIL method). This means we multiply each term in the first binomial by each term in the second binomial:
( x + 2 ) ( x + 3 ) = x ⋅ x + x ⋅ 3 + 2 ⋅ x + 2 ⋅ 3 = x 2 + 3 x + 2 x + 6
Simplifying the Expression Now, we simplify the expression by combining like terms: x 2 + 3 x + 2 x + 6 = x 2 + 5 x + 6 .
Final Multiplication and Result Finally, we multiply the resulting trinomial by the constant factor 6: 6 ( x 2 + 5 x + 6 ) = 6 ⋅ x 2 + 6 ⋅ 5 x + 6 ⋅ 6 = 6 x 2 + 30 x + 36 . Therefore, the simplified polynomial function is P ( x ) = 6 x 2 + 30 x + 36 .
Examples
Polynomial functions are used in various real-world applications, such as modeling the trajectory of a ball, designing roller coasters, or even predicting population growth. For example, if you throw a ball, its height over time can be modeled by a quadratic polynomial. Understanding how to manipulate and simplify these functions is crucial in many scientific and engineering fields. In this case, we simplified a polynomial function by multiplying its factors, which is a fundamental skill in algebra.
