- Multiply the binomials \((3x^3-2)(4x^2+5x)\) using the distributive property to get \(12x^5 + 15x^4 - 8x^2 - 10x\).
- Multiply this result by the monomial \(x^4\).
- Apply the exponent rule \(x^m \cdot x^n = x^{m+n}\) to each term.
- Obtain the final expression \(12x^9 + 15x^8 - 8x^6 - 10x^5\), which matches option \(\boxed{d}\).
Explanation
- Problem Analysis and Strategy
The objective is to simplify the algebraic expression \(\left(x^4\right)\left(3 x^3-2\right)\left(4 x^2+5 x\right)\) and determine which of the provided options it matches. The expression involves the multiplication of a monomial by two binomials. We can use the associative property of multiplication to group the factors and the distributive property to expand the products.
- Multiplying the Binomials
First, we multiply the two binomials: \(\left(3 x^3-2\right)\) and \(\left(4 x^2+5 x\right)\). Using the distributive property (often referred to as the FOIL method for binomials), we have:
\[\left(3 x^3-2\right)\left(4 x^2+5 x\right) = (3x^3)(4x^2) + (3x^3)(5x) + (-2)(4x^2) + (-2)(5x)\]
Applying the exponent rule \(x^m \cdot x^n = x^{m+n}\), we get:
\[(3x^3)(4x^2) = 12x^{3+2} = 12x^5\]
\[(3x^3)(5x) = 15x^{3+1} = 15x^4\]
\[(-2)(4x^2) = -8x^2\]
\[(-2)(5x) = -10x\]
Combining these, the product of the binomials is \(12x^5 + 15x^4 - 8x^2 - 10x\).
- Multiplying by the Monomial
Next, we multiply this result by the monomial \(\left(x^4\right)\) using the distributive property:
\[x^4 \left(12x^5 + 15x^4 - 8x^2 - 10x\right)\]
Multiplying each term by \(x^4\):
\[(x^4)(12x^5) + (x^4)(15x^4) + (x^4)(-8x^2) + (x^4)(-10x)\]
Using the exponent rule again (\(x^m \cdot x^n = x^{m+n}\)):
\[12x^{4+5} + 15x^{4+4} - 8x^{4+2} - 10x^{4+1}\]
\[= 12x^9 + 15x^8 - 8x^6 - 10x^5\]
- Conclusion
Comparing our simplified expression \(12x^9 + 15x^8 - 8x^6 - 10x^5\) to the given options:
A\(12 x^{24}+15 x^{12}-8 x^8-10 x^4\)
B\(12 x^{24}-10 x^4\)
C\(12 x^9-10 x^5\)
D\(12 x^9+15 x^8-8 x^6-10 x^5\)
Our result matches option d. Thus, the simplified form is \(12x^9 + 15x^8 - 8x^6 - 10x^5\), which corresponds to option \(\boxed{d}\).
Examples
Polynomial multiplication is used in various fields, such as physics and computer graphics. For example, if you are modeling the volume of a geometric shape where the dimensions are given as functions of a variable \(x\), such as length \(l(x) = x^4\), width \(w(x) = 3x^3 - 2\), and height \(h(x) = 4x^2 + 5x\), the total volume is found by multiplying these expressions: \(V(x) = l(x) \cdot w(x) \cdot h(x)\). Mastering polynomial expansion allows you to represent such complex relationships as a single, easy-to-analyze polynomial.