Find each product and write the result in standard form. (−5 + 4i)(3 + i)

검증된 단계별 안내

Recall that to multiply two complex numbers in the form $(a + bi)(c + di)$, you use the distributive property (FOIL method): multiply each term in the first complex number by each term in the second complex number.

Apply the distributive property: $(−5 + 4i)(3 + i) = (−5)(3) + (−5)(i) + (4i)(3) + (4i)(i)$.

Calculate each product separately: $(−5)(3) = −15$, $(−5)(i) = −5i$, $(4i)(3) = 12i$, and $(4i)(i) = 4i^2$.

Remember that $i^2 = -1$, so replace \$4i^2$ with $4(-1) = -4$.

Combine the real parts and the imaginary parts: real parts are $-15$ and $-4$, imaginary parts are $-5i$ and \$12i$. Write the final expression in the form $a + bi$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Complex Number Multiplication

Multiplying complex numbers involves using the distributive property (FOIL method) to expand the product of two binomials. Each term is multiplied, remembering that i² = -1, which simplifies the expression into a standard form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary coefficient. After multiplication, the result should be simplified and rearranged to clearly separate real and imaginary parts.

Imaginary Unit Properties

The imaginary unit i is defined such that i² = -1. This property is essential when simplifying products involving i, as it converts powers of i into real numbers, allowing the expression to be written in standard form.