Answered

Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

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.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.