Question

what is the remainder when (4x³ + 2x² - 18x + 38) ÷ (x + 3)?
2
12
96
110

Explanation:

Step1: Apply Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( (x - a) \), the remainder is \( f(a) \). For \( (x + 3) \), we can write it as \( (x - (-3)) \), so \( a = -3 \).

Step2: Substitute \( x = -3 \) into \( f(x) = 4x^3 + 2x^2 - 18x + 38 \)

Calculate each term:

• \( 4(-3)^3 = 4(-27) = -108 \)
• \( 2(-3)^2 = 2(9) = 18 \)
• \( -18(-3) = 54 \)
• The constant term is \( 38 \)

Now sum these terms: \( -108 + 18 + 54 + 38 \)
First, \( -108 + 18 = -90 \)
Then, \( -90 + 54 = -36 \)
Finally, \( -36 + 38 = 2 \)? Wait, no, wait, let's recalculate:
Wait, \( 4(-3)^3 = 4*(-27) = -108 \)
\( 2(-3)^2 = 2*9 = 18 \)
\( -18*(-3) = 54 \)
\( 38 \)
Now add all together: \( -108 + 18 = -90 \); \( -90 + 54 = -36 \); \( -36 + 38 = 2 \)? But that's not matching. Wait, maybe I made a mistake. Wait, no, let's check again. Wait, the polynomial is \( 4x^3 + 2x^2 - 18x + 38 \). Let's substitute \( x = -3 \):

\( 4(-3)^3 = 4(-27) = -108 \)

\( 2(-3)^2 = 29 = 18 \)

\( -18*(-3) = 54 \)

\( +38 \)

Now sum: \( -108 + 18 = -90 \); \( -90 + 54 = -36 \); \( -36 + 38 = 2 \). Wait, but the options have 2, 12, 96, 110. Wait, maybe I messed up the sign. Wait, \( x + 3 = 0 \) implies \( x = -3 \), correct. Wait, let's recalculate:

\( 4x^3 = 4(-3)^3 = 4(-27) = -108 \)

\( 2x^2 = 2(-3)^2 = 29 = 18 \)

\( -18x = -18*(-3) = 54 \)

\( +38 \)

So total: \( -108 + 18 + 54 + 38 \). Let's add 18 + 54 = 72; 72 + 38 = 110; 110 - 108 = 2. Wait, that's 2. But wait, maybe I made a mistake. Wait, no, the Remainder Theorem is correct. Wait, let's do polynomial long division to check.

Divide \( 4x^3 + 2x^2 - 18x + 38 \) by \( x + 3 \).

First term: \( 4x^3 / x = 4x^2 \). Multiply \( (x + 3) \) by \( 4x^2 \): \( 4x^3 + 12x^2 \). Subtract from the original polynomial:

\( (4x^3 + 2x^2 - 18x + 38) - (4x^3 + 12x^2) = -10x^2 - 18x + 38 \)

Next term: \( -10x^2 / x = -10x \). Multiply \( (x + 3) \) by \( -10x \): \( -10x^2 - 30x \). Subtract:

\( (-10x^2 - 18x + 38) - (-10x^2 - 30x) = 12x + 38 \)

Next term: \( 12x / x = 12 \). Multiply \( (x + 3) \) by \( 12 \): \( 12x + 36 \). Subtract:

\( (12x + 38) - (12x + 36) = 2 \). So the remainder is 2. Wait, but the options have 2 as the first option. So maybe the initial calculation was correct. Wait, but when I did the Remainder Theorem, I got 2. So the remainder is 2.

Answer:

2 (the first option, A. 2 if we consider the options as A. 2, B. 12, C. 96, D. 110)