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Mathematics

12. When the polynomials [tex]\(a x^3 + 5 x^2 + 3 x + 4\)[/tex] and [tex]\(3 x^3 + 9 x^2 + a x - 6\)[/tex] are divided by [tex]\(x+3\)[/tex], the remainders are [tex]\(A\)[/tex] and [tex]\(B\)[/tex] respectively. Find the values of [tex]\(a\)[/tex] if [tex]\(A + B = 4\)[/tex].

13. The remainder when [tex]\(a x^3 + b x^2 + 2 x + 3\)[/tex] is divided by [tex]\(x-1\)[/tex] is twice that when it is divided by [tex]\(x+1\)[/tex]. Show that [tex]\(b = 3a + 3\)[/tex].

14. The remainder when [tex]\(2 x^3 + k x^2 + 7\)[/tex] is divided by [tex]\(x-2\)[/tex] is half the remainder when the same expression is divided by [tex]\(2x-1\)[/tex]. Find the value of [tex]\(k\)[/tex].

15. The remainder when [tex]\(x^4 + 3 x^2 - 2 x + 2\)[/tex] is divided by [tex]\(x + a\)[/tex] is the square of the remainder when [tex]\(x^2 - 3\)[/tex] is divided by [tex]\(x + a\)[/tex]. Calculate the possible values of [tex]\(a\)[/tex].

16. The expression [tex]\(a x^3 - x^2 + b x - 1\)[/tex] leaves remainders of -33 and 77 when divided by [tex]\(x+2\)[/tex] and [tex]\(x-3\)[/tex] respectively. Find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] and the remainder when divided by [tex]\(x-2\)[/tex].

Sagot :

GinnyAnswer
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.
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