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Sagot :
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.
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.
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