- Distribute each term of the first polynomial over the second: \((-3x^5)(7x^2 - 2x + 6) + (-4x^4)(7x^2 - 2x + 6)\).
- Perform individual multiplications: \(-21x^7 + 6x^6 - 18x^5 - 28x^6 + 8x^5 - 24x^4\).
- Simplify by combining like terms (terms with the same powers of \(x\)).
- The final expression is \(\boxed{-21x^7 - 22x^6 - 10x^5 - 24x^4}\).
Explanation
- Setting up the Multiplication
To find the product of the polynomials \((-3x^5 - 4x^4)\) and \((7x^2 - 2x + 6)\), we apply the distributive property, which states that for any expressions \(a, b, c,\) and \(d\), \((a + b)(c + d + e) = ac + ad + ae + bc + bd + be.\) In this case, we distribute each term of the first polynomial across the terms of the second polynomial.
- Calculating Individual Terms
We multiply each term of \((-3x^5 - 4x^4)\) by each term of \((7x^2 - 2x + 6)\):
- \((-3x^5) \cdot (7x^2) = -21x^{5+2} = -21x^7\)
- \((-3x^5) \cdot (-2x) = 6x^{5+1} = 6x^6\)
- \((-3x^5) \cdot (6) = -18x^5\)
- \((-4x^4) \cdot (7x^2) = -28x^{4+2} = -28x^6\)
- \((-4x^4) \cdot (-2x) = 8x^{4+1} = 8x^5\)
- \((-4x^4) \cdot (6) = -24x^4\)
- Combining Like Terms and Final Result
Now, we gather all the intermediate results and combine like terms (those with the same exponent):
\(-21x^7 + 6x^6 - 18x^5 - 28x^6 + 8x^5 - 24x^4\)
Grouping the like terms:
- \(x^7\) term: \(-21x^7\)
- \(x^6\) terms: \(6x^6 - 28x^6 = -22x^6\)
- \(x^5\) terms: \(-18x^5 + 8x^5 = -10x^5\)
- \(x^4\) term: \(-24x^4\)
Combining them gives the final expanded polynomial: \(-21x^7 - 22x^6 - 10x^5 - 24x^4\).
Examples
Polynomial multiplication is used frequently in engineering and physics to model complex systems. For instance, if you are calculating the total area of a land plot where dimensions are defined by variable expressions, or determining the trajectory of an object where velocity and time are expressed as functions, you end up multiplying polynomials to find the final result. Understanding this process allows you to simplify these models into a single, manageable expression.