Expand And Fully Simplify The Expression $(2x+5)(4x-3)(5x-4)$ To Give An Expression Of The Form $ax^3 + Bx^2 + Cx + D$. Determine The Values Of A A A , B B B , C C C , And D D D .

Introduction

In this article, we will expand and simplify the given expression $(2x+5)(4x-3)(5x-4)$ to give an expression of the form $ax^3 + bx^2 + cx + d$. This involves using the distributive property to multiply the three binomials together and then combining like terms to obtain the final expression.

Step 1: Multiply the First Two Binomials

To expand the given expression, we start by multiplying the first two binomials, $(2x+5)$ and $(4x-3)$. We can use the distributive property to multiply each term in the first binomial by each term in the second binomial.

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

binomial1 = 2x + 5
binomial2 = 4x - 3

product1 = sp.expand(binomial1 * binomial2)
print(product1)

This will output the product of the first two binomials, which is $8x^2 - 6x + 20x - 15 = 8x^2 + 14x - 15$.

Step 2: Multiply the Result by the Third Binomial

Next, we multiply the result from Step 1 by the third binomial, $(5x-4)$. Again, we use the distributive property to multiply each term in the product from Step 1 by each term in the third binomial.

# Define the third binomial
binomial3 = 5*x - 4

product2 = sp.expand(product1 * binomial3)
print(product2)

This will output the final product, which is $40x^3 - 32x^2 + 70x^2 - 56x - 75x + 60 = 40x^3 + 38x^2 - 131x + 60$.

Step 3: Determine the Values of $a$, $b$, $c$, and $d$

Now that we have the final product, we can determine the values of $a$, $b$, $c$, and $d$ by comparing the product with the general form $ax^3 + bx^2 + cx + d$.

• $a$ is the coefficient of the term with the highest degree, which is $40x^3$. Therefore, $a = 40$.
• $b$ is the coefficient of the term with the second-highest degree, which is $38x^2$. Therefore, $b = 38$.
• $c$ is the coefficient of the term with the lowest degree, which is $-131x$. Therefore, $c = -131$.
• $d$ is the constant term, which is $60$. Therefore, $d = 60$.

Conclusion

In this article, we expanded and simplified the given expression $(2x+5)(4x-3)(5x-4)$ to give an expression of the form $ax^3 + bx^2 + cx + d$. We used the distributive property to multiply the three binomials together and then combined like terms to obtain the final expression. Finally, we determined the values of $a$, $b$, $c$, and $d$ by comparing the product with the general form.

Final Answer

Introduction

In our previous article, we expanded and simplified the given expression $(2x+5)(4x-3)(5x-4)$ to give an expression of the form $ax^3 + bx^2 + cx + d$. We used the distributive property to multiply the three binomials together and then combined like terms to obtain the final expression. In this article, we will answer some frequently asked questions related to expanding and simplifying the expression of a cubic polynomial.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to multiply a single term by two or more terms. It states that for any numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

This property is used extensively in algebra to simplify expressions and solve equations.

Q: How do I multiply two binomials together?

A: To multiply two binomials together, we use the distributive property to multiply each term in the first binomial by each term in the second binomial. For example, to multiply $(2x+5)$ and $(4x-3)$, we would multiply each term in the first binomial by each term in the second binomial:

$$(2x+5)(4x-3) = 2x(4x) + 2x(-3) + 5(4x) + 5(-3)$$

This would give us the product $8x^2 - 6x + 20x - 15 = 8x^2 + 14x - 15$.

Q: How do I multiply a binomial by a trinomial?

A: To multiply a binomial by a trinomial, we use the distributive property to multiply each term in the binomial by each term in the trinomial. For example, to multiply $(2x+5)$ and $(4x-3)(5x-4)$, we would multiply each term in the binomial by each term in the trinomial:

$$(2x+5)(4x-3)(5x-4) = (2x+5)(8x^2 + 14x - 15)(5x-4)$$

This would give us the product $40x^3 + 38x^2 - 131x + 60$.

Q: What is the general form of a cubic polynomial?

A: The general form of a cubic polynomial is $ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants. This form is used to represent any cubic polynomial, and it is the form that we used to represent the expression $(2x+5)(4x-3)(5x-4)$.

Q: How do I determine the values of $a$, $b$, $c$, and $d$?

A: To determine the values of $a$, $b$, $c$, and $d$, we compare the given expression with the general form $ax^3 + bx^2 + cx + d$. We identify the coefficients of each term in the given expression and match them with the corresponding coefficients in the general form. For example, in the expression $(2x+5)(4x-3)(5x-4)$, we have $a = 40$, $b = 38$, $c = -131$, and $d = 60$.

Conclusion

In this article, we answered some frequently asked questions related to expanding and simplifying the expression of a cubic polynomial. We discussed the distributive property, how to multiply two binomials together, how to multiply a binomial by a trinomial, the general form of a cubic polynomial, and how to determine the values of $a$, $b$, $c$, and $d$. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the topic.

Final Answer

The final answer is $\boxed{40x^3 + 38x^2 - 131x + 60}$.