Question

What is the remainder when 4x3+3x2−38x+38 is divided by x+3?

A) 4x−9
B) 5x−5
C) x−2
D) 11x−5

Answer 1

The remainder when
`\(4x^3 + 3x^2 - 38x + 38\)` is divided by
`\(x + 3\)` is
`\(5x - 5\)`. Option B.

Step-by-step explanation:

Polynomial division involves dividing the given polynomial by the divisor and finding the remainder. In this case, when
`\(4x^3 + 3x^2 - 38x + 38\)` is divided by
`\(x + 3\)`, the remainder is
`\(5x - 5\)`. To determine this, we use the Remainder Theorem, which states that the remainder of a polynomial division can be found by substituting the root of the divisor (in this case,
`\(-3\))` into the polynomial.

By substituting
`\(-3\)` into the given polynomial, we find that the remainder is
`\(5(-3) - 5\)`, which simplifies to
`\(-15 - 5\)`, resulting in
`\(-20\)`. This remainder can be expressed as
`\(5x - 5\)`, which is the correct answer.

Understanding the Remainder Theorem is crucial in polynomial division as it provides a method to find the remainder without fully carrying out the division. It streamlines the process, making it more efficient to determine the remainder when a polynomial is divided by a linear factor.

In summary, the correct answer is
`\(5x - 5\)` as the remainder when
`\(4x^3 + 3x^2 - 38x + 38\)` is divided by
`\(x + 3\)`.