Question
Select the product \( (4+i)(-2-i) \) in the standard form \( a+b i \). a.) \( -8-i^{2} \) b.) \( 8-i^{2} \) c.) \( 7-6 i \) d.) \( -7-6 i \)
Ask by Olson Hardy. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The product is \( -7-6i \).
Solution
Calculate or simplify the expression \( (4+i)(-2-i) \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(4+i\right)\left(-2-i\right)\)
- step1: Apply the distributive property:
\(4\left(-2\right)+4\left(-i\right)+i\left(-2\right)+i\left(-i\right)\)
- step2: Multiply the numbers:
\(-8+4\left(-i\right)+i\left(-2\right)+i\left(-i\right)\)
- step3: Multiply the numbers:
\(-8-4i+i\left(-2\right)+i\left(-i\right)\)
- step4: Multiply the numbers:
\(-8-4i-2i+i\left(-i\right)\)
- step5: Multiply the numbers:
\(-8-4i-2i+1\)
- step6: Calculate:
\(-7-4i-2i\)
- step7: Multiply the numbers:
\(-7-6i\)
The product \( (4+i)(-2-i) \) in the standard form \( a+bi \) is \( -7-6i \).
Therefore, the correct answer is d.) \( -7-6i \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the product \( (4+i)(-2-i) \), we can use the distributive property (also known as the FOIL method for binomials): \[ (4+i)(-2-i) = 4 \cdot (-2) + 4 \cdot (-i) + i \cdot (-2) + i \cdot (-i) \] Calculating each term: - \( 4 \cdot (-2) = -8 \) - \( 4 \cdot (-i) = -4i \) - \( i \cdot (-2) = -2i \) - \( i \cdot (-i) = -i^2 = -(-1) = 1 \) (since \( i^2 = -1 \)) Now, add all the results together: \[ -8 - 4i - 2i + 1 = -8 + 1 - 6i = -7 - 6i \] Thus, the product in standard form \( a + bi \) is: \(-7 - 6i\) The correct answer is: d.) \( -7 - 6 i \)
