Expand and simplify the following expression: (6 + d)(2d^2 - d + 7)
Understand the Problem
The question requires expanding the product of two polynomial expressions. We need to multiply each term in the first polynomial (6 + d) by each term in the second polynomial (2d^2 - d + 7) and then simplify the resulting expression by combining like terms.
Answer
$2d^3 + 11d^2 + d + 42$
Answer for screen readers
$2d^3 + 11d^2 + d + 42$
Steps to Solve
- Distribute 6 into the second polynomial
Multiply 6 by each term in $(2d^2 - d + 7)$:
$6 * (2d^2 - d + 7) = 12d^2 - 6d + 42$
- Distribute d into the second polynomial
Multiply $d$ by each term in $(2d^2 - d + 7)$:
$d * (2d^2 - d + 7) = 2d^3 - d^2 + 7d$
- Combine the results
Add the two resulting polynomials from steps 1 and 2:
$(12d^2 - 6d + 42) + (2d^3 - d^2 + 7d) = 2d^3 + (12d^2 - d^2) + (-6d + 7d) + 42$
- Simplify by combining like terms
Combine the $d^2$ terms and the $d$ terms:
$2d^3 + (12d^2 - d^2) + (-6d + 7d) + 42 = 2d^3 + 11d^2 + d + 42$
$2d^3 + 11d^2 + d + 42$
More Information
The expanded form of the given polynomial is $2d^3 + 11d^2 + d + 42$. There are no further simplifications possible.
Tips
A common mistake is to forget to distribute either the 6 or the $d$ across all three terms of the second polynomial. Another common mistake is in combining the like terms, specifically with incorrect signs. For example, a sign error during the addition of $-6d$ and $7d$. Carefully check each term after distribution and combination.
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