To multiply the polynomials \((2x - 7)(3x^2 + 7x - 8)\), we'll use the distributive property.
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Distributing \(2x\) to each term in \(3x^2 + 7x - 8\): \[ 2x \cdot 3x^2 = 6x^3 \] \[ 2x \cdot 7x = 14x^2 \] \[ 2x \cdot (-8) = -16x \]
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Distributing \(-7\) to each term in \(3x^2 + 7x - 8\): \[ -7 \cdot 3x^2 = -21x^2 \] \[ -7 \cdot 7x = -49x \] \[ -7 \cdot (-8) = 56 \]
Now, combine all these results:
- From \(2x\) we got: \(6x^3 + 14x^2 - 16x\)
- From \(-7\) we got: \(-21x^2 - 49x + 56\)
Combine the like terms:
- \(6x^3\) (only one term)
- \(14x^2 - 21x^2 = -7x^2\)
- \(-16x - 49x = -65x\)
- \(56\) (only one term)
Putting it all together, the result of the multiplication is: \[ 6x^3 - 7x^2 - 65x + 56 \]
Now, we can identify the coefficients for each power of \(x\):
- Coefficient of \(x^3\) is 6
- Coefficient of \(x^2\) is -7
- Coefficient of \(x\) is -65
- Constant term (coefficient of \(x^0\)) is 56
Putting the answers neatly:
- The value of the coefficient of \(x^3\) is 6
- The value of the coefficient of \(x^2\) is -7
- The value of the coefficient of \(x\) is -65
- The value of the coefficient of the constant is 56
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