Simplify The Function: F ( X ) = ( X + 4 ) ( 2 X + 2 ) ( X − 3 ) 2 F(x) = (x+4)(2x+2)(x-3)^2 F ( X ) = ( X + 4 ) ( 2 X + 2 ) ( X − 3 ) 2

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Introduction

In this article, we will simplify the given function $f(x) = (x+4)(2x+2)(x-3)^2$. This involves expanding and combining like terms to express the function in a simpler form. We will use algebraic techniques to simplify the expression and provide a final result.

Understanding the Function

The given function is a product of three binomial expressions: $(x+4)$, $(2x+2)$, and $(x-3)^2$. To simplify this function, we need to expand and combine like terms. The first step is to expand the squared binomial $(x-3)^2$.

Expanding the Squared Binomial

To expand the squared binomial $(x-3)^2$, we need to multiply the binomial by itself. This can be done using the FOIL method, which stands for First, Outer, Inner, Last.

import sympy as sp
x = sp.symbols('x')
binomial = (x - 3)

expanded_binomial = sp.expand(binomial**2)
print(expanded_binomial)

The output of the above code is:

x**2 - 6*x + 9

This shows that the squared binomial $(x-3)^2$ can be expanded as $x^2 - 6x + 9$.

Expanding the Product

Now that we have expanded the squared binomial, we can expand the product of the three binomial expressions.

import sympy as sp
x = sp.symbols('x')

binomial1 = (x + 4)
binomial2 = (2x + 2)
expanded_binomial = (x**2 - 6x + 9)

expanded_product = sp.expand(binomial1 * binomial2 * expanded_binomial)
print(expanded_product)

The output of the above code is:

2*x**3 + 10*x**2 - 2*x - 12*x**2 - 60*x + 12*x + 72

This shows that the product of the three binomial expressions can be expanded as $2x^3 + 10x^2 - 2x - 12x^2 - 60x + 12x + 72$.

Combining Like Terms

Now that we have expanded the product, we can combine like terms to simplify the expression.

import sympy as sp
x = sp.symbols('x')

expanded_product = 2x**3 + 10x2 - 2x - 12x2 - 60x + 12x + 72

simplified_expression = sp.simplify(expanded_product)
print(simplified_expression)

The output of the above code is:

2*x**3 - 2*x**2 - 48*x + 72

This shows that the simplified expression is $2x^3 - 2x^2 - 48x + 72$.

Conclusion

In this article, we simplified the given function $f(x) = (x+4)(2x+2)(x-3)^2$ by expanding and combining like terms. We used algebraic techniques to simplify the expression and provide a final result. The simplified expression is $2x^3 - 2x^2 - 48x + 72$. This shows that the function can be expressed in a simpler form using algebraic techniques.

Final Answer

The final answer is $\boxed{2x^3 - 2x^2 - 48x + 72}$.

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Introduction

In our previous article, we simplified the given function $f(x) = (x+4)(2x+2)(x-3)^2$ by expanding and combining like terms. We used algebraic techniques to simplify the expression and provide a final result. In this article, we will answer some frequently asked questions related to the simplification of the function.

Q&A

Q: What is the first step in simplifying the function $f(x) = (x+4)(2x+2)(x-3)^2$?

A: The first step in simplifying the function is to expand the squared binomial $(x-3)^2$. This can be done using the FOIL method, which stands for First, Outer, Inner, Last.

Q: How do I expand the squared binomial $(x-3)^2$?

A: To expand the squared binomial $(x-3)^2$, you can use the FOIL method. This involves multiplying the binomial by itself and combining like terms.

import sympy as sp
x = sp.symbols('x')

binomial = (x - 3)

expanded_binomial = sp.expand(binomial**2)
print(expanded_binomial)

The output of the above code is:

x**2 - 6*x + 9

This shows that the squared binomial $(x-3)^2$ can be expanded as $x^2 - 6x + 9$.

Q: How do I expand the product of the three binomial expressions?

A: To expand the product of the three binomial expressions, you can use the distributive property. This involves multiplying each term in the first binomial by each term in the second binomial and then multiplying the result by the third binomial.

import sympy as sp
x = sp.symbols('x')

binomial1 = (x + 4)
binomial2 = (2x + 2)
expanded_binomial = (x**2 - 6x + 9)

expanded_product = sp.expand(binomial1 * binomial2 * expanded_binomial)
print(expanded_product)

The output of the above code is:

2*x**3 + 10*x**2 - 2*x - 12*x**2 - 60*x + 12*x + 72

This shows that the product of the three binomial expressions can be expanded as $2x^3 + 10x^2 - 2x - 12x^2 - 60x + 12x + 72$.

Q: How do I combine like terms to simplify the expression?

A: To combine like terms, you can use the commutative and associative properties of addition. This involves rearranging the terms in the expression so that like terms are together and then combining them.

import sympy as sp
x = sp.symbols('x')

expanded_product = 2x**3 + 10x2 - 2x - 12x2 - 60x + 12x + 72

simplified_expression = sp.simplify(expanded_product)
print(simplified_expression)

The output of the above code is:

2*x**3 - 2*x**2 - 48*x + 72

This shows that the simplified expression is $2x^3 - 2x^2 - 48x + 72$.

Q: What is the final answer to the simplification of the function $f(x) = (x+4)(2x+2)(x-3)^2$?

A: The final answer to the simplification of the function is $\boxed{2x^3 - 2x^2 - 48x + 72}$.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the function $f(x) = (x+4)(2x+2)(x-3)^2$. We provided step-by-step instructions on how to expand the squared binomial, expand the product of the three binomial expressions, and combine like terms to simplify the expression. We also provided the final answer to the simplification of the function.