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Multiply the following polynomials and write the resulting polynomials in descending order.

(2x−7)(3x2+7x−8)
(4 points)
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The value of the coefficient of the x3
is Response area

The value of the coefficient of the x2
is Response area
The value of the coefficient of the x
is Response area
The value of the coefficient of the constant is Response area
The option "-6" (5 of 11) has been selected. Press tab to choose a response area, and spacebar to insert it. Press escape to cancel.
6,-7,-65, 56, -6,7,-56,0,-11,12,78

1 answer

To multiply the polynomials \( (2x - 7)(3x^2 + 7x - 8) \), we will use the distributive property (also known as the FOIL method for binomials).

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

    • \( 2x \cdot 3x^2 = 6x^3 \)
    • \( 2x \cdot 7x = 14x^2 \)
    • \( 2x \cdot (-8) = -16x \)

  2. Multiply \(-7\) by each term in the second polynomial:

    • \(-7 \cdot 3x^2 = -21x^2\)
    • \(-7 \cdot 7x = -49x\)
    • \(-7 \cdot (-8) = 56\)

  3. Now, combine all these results:

    • \( 6x^3 + 14x^2 - 16x - 21x^2 - 49x + 56 \)

  4. Combine like terms:

    • For \(x^2\): \(14x^2 - 21x^2 = -7x^2\)
    • For \(x\): \(-16x - 49x = -65x\)

So the resulting polynomial is: \[ 6x^3 - 7x^2 - 65x + 56 \]

Now, to fill in the coefficient responses:

  • The value of the coefficient of the \(x^3\) term is 6.
  • The value of the coefficient of the \(x^2\) term is -7.
  • The value of the coefficient of the \(x\) term is -65.
  • The value of the coefficient of the constant is 56.

So the final answers are:

  • Coefficient of \(x^3\): 6
  • Coefficient of \(x^2\): -7
  • Coefficient of \(x\): -65
  • Coefficient of constant: 56