Let's solve the given problem step-by-step.
Given polynomials: [tex]\( a x^3 + 5 x^2 + 3 x + 4 \)[/tex] and [tex]\( 3 x^3 + 9 x^2 + a x - 6 \)[/tex]. When these polynomials are divided by [tex]\( x+3 \)[/tex], the remainders are [tex]\( A \)[/tex] and [tex]\( B \)[/tex] respectively.
We are also given that [tex]\( A + B = 4 \)[/tex].
When a polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( x-c \)[/tex], the remainder is [tex]\( P(c) \)[/tex].
1. Find the remainder [tex]\( A \)[/tex] when [tex]\( a x^3 + 5 x^2 + 3 x + 4 \)[/tex] is divided by [tex]\( x+3 \)[/tex]:
To find [tex]\( A \)[/tex], we substitute [tex]\( x = -3 \)[/tex] into the polynomial [tex]\( a x^3 + 5 x^2 + 3 x + 4 \)[/tex].
[tex]\[
A = a(-3)^3 + 5(-3)^2 + 3(-3) + 4
\][/tex]
Simplifying this:
[tex]\[
A = a(-27) + 5(9) - 9 + 4
\][/tex]
[tex]\[
A = -27a + 45 - 9 + 4
\][/tex]
[tex]\[
A = -27a + 40
\][/tex]
2. Find the remainder [tex]\( B \)[/tex] when [tex]\( 3 x^3 + 9 x^2 + a x - 6 \)[/tex] is divided by [tex]\( x+3 \)[/tex]:
To find [tex]\( B \)[/tex], we substitute [tex]\( x = -3 \)[/tex] into the polynomial [tex]\( 3 x^3 + 9 x^2 + a x - 6 \)[/tex].
[tex]\[
B = 3(-3)^3 + 9(-3)^2 + a(-3) - 6
\][/tex]
Simplifying this:
[tex]\[
B = 3(-27) + 9(9) - 3a - 6
\][/tex]
[tex]\[
B = -81 + 81 - 3a - 6
\][/tex]
[tex]\[
B = -3a - 6
\][/tex]
3. Given [tex]\( A + B = 4 \)[/tex], substitute the expressions for [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A + B = (-27a + 40) + (-3a - 6) = 4
\][/tex]
Combine like terms:
[tex]\[
-27a - 3a + 40 - 6 = 4
\][/tex]
Simplify:
[tex]\[
-30a + 34 = 4
\][/tex]
Subtract 34 from both sides:
[tex]\[
-30a = -30
\][/tex]
Divide by -30:
[tex]\[
a = 1
\][/tex]
By our calculations, we've found that the value of [tex]\( a \)[/tex] is 1.
