Question

What is the product of the following two polynomials?(2x3+3x2)(4x4−5x3−6x2) (1 point)Responses8x7+2x6−27x5−18x48 x superscript 7 baseline plus 2 x superscript 6 baseline minus 27 x superscript 5 baseline minus 18 x superscript 4 baseline8x7+2x6−3x5−18x48 x superscript 7 baseline plus 2 x superscript 6 baseline minus 3 x superscript 5 baseline minus 18 x superscript 4 baseline8x7+22x6−27x5+18x48 x superscript 7 baseline plus 22 x superscript 6 baseline minus 27 x superscript 5 baseline plus 18 x superscript 4 baseline8x7−22x6−27x5−18x48 x superscript 7 baseline minus 22 x superscript 6 baseline minus 27 x superscript 5 baseline minus 18 x superscript 4 baseline

Answer 1

To find the product of the two polynomials \( (2x^3 + 3x^2) \) and \( (4x^4 - 5x^3 - 6x^2) \), we will use the distributive property (also known as the FOIL method for binomials) to multiply every term in the first polynomial by every term in the second polynomial.

Let's break it down:

1. Multiply \( 2x^3 \) by each term in the second polynomial:

   • \( 2x^3 \cdot 4x^4 = 8x^7 \)
   • \( 2x^3 \cdot (-5x^3) = -10x^6 \)
   • \( 2x^3 \cdot (-6x^2) = -12x^5 \)

2. Multiply \( 3x^2 \) by each term in the second polynomial:

   • \( 3x^2 \cdot 4x^4 = 12x^6 \)
   • \( 3x^2 \cdot (-5x^3) = -15x^5 \)
   • \( 3x^2 \cdot (-6x^2) = -18x^4 \)

Now, combine all the results:

Combining like terms, we have:

• For \( x^7 \): \( 8x^7 \)
• For \( x^6 \): \( -10x^6 + 12x^6 = 2x^6 \)
• For \( x^5 \): \( -12x^5 - 15x^5 = -27x^5 \)
• For \( x^4 \): \( -18x^4 \)

Putting it all together, the resulting polynomial is:

\[ 8x^7 + 2x^6 - 27x^5 - 18x^4 \]

Thus, the correct response is:

8x^7 + 2x^6 - 27x^5 - 18x^4