1. Given $$ 3 x^{3}+8 x^{2}-1 \equiv(x+5)(x-2) Q(x)+A x+B $$, where $$ Q(x) $$ is a polynomial. Find $$ \begin{array}{ll} ext { (i) } Q(x) ext {, } \ ext { (ii) the values of } A ext { and } B ext {. } \ ext { [Answer Key] } \ \begin{array}{ll} ext { (i) } 3 x-1 & ext { (ii) } A=33, \quad B=-11\end{array}\end{array} $$.

Question

1. Given $$ 3 x^{3}+8 x^{2}-1 \equiv(x+5)(x-2) Q(x)+A x+B $$, where $$ Q(x) $$ is a polynomial. Find $$ \begin{array}{ll}	ext { (i) } Q(x) 	ext {, } \ 	ext { (ii) the values of } A 	ext { and } B 	ext {. } \ 	ext { [Answer Key] } \ \begin{array}{ll}	ext { (i) } 3 x-1 & 	ext { (ii) } A=33, \quad B=-11\end{array}\end{array}> $$.

UpStudy AI · Accepted Answer

We write the given identity as

$$
3x^3+8x^2-1 = (x+5)(x-2) Q(x) + Ax+B.
$$

Since $$(x+5)(x-2)$$ is a quadratic, for the degree of the cubic $$3x^3+8x^2-1$$ to be maintained, $$Q(x)$$ must be a linear polynomial. Let

$$
Q(x)=ax+b.
$$

First, compute the product:

$$
(x+5)(x-2)= x^2+3x-10.
$$

Then

$$
\begin{aligned}
(x^2+3x-10)(ax+b) & = ax^3 + bx^2 + 3ax^2 + 3bx - 10ax - 10b \
& = ax^3 + (b+3a)x^2 + (3b-10a)x - 10b.
\end{aligned}
$$

Including the remainder $$Ax+B$$, the entire right-hand side becomes

$$
ax^3 + (b+3a)x^2 + \bigl(3b-10a+ A\bigr)x + \bigl(-10b+B\bigr).
$$

This must equal the left-hand side term by term:

$$
3x^3+8x^2+0x-1.
$$

Equate coefficients of like powers of $$x$$:

1. Coefficient of $$x^3$$:
   $$
   a=3.
   $$

2. Coefficient of $$x^2$$:
   $$
   b+3a=8 \quad \Longrightarrow \quad b+3(3)=8 \quad \Longrightarrow \quad b+9=8 \quad \Longrightarrow \quad b=-1.
   $$

3. Coefficient of $$x$$:
   $$
   3b-10a+ A=0 \quad \Longrightarrow \quad 3(-1)-10(3)+A=0 \quad \Longrightarrow \quad -3-30+A=0 \quad \Longrightarrow \quad A=33.
   $$

4. Constant term:
   $$
   -10b+B=-1 \quad \Longrightarrow \quad -10(-1)+B=-1 \quad \Longrightarrow \quad 10+B=-1 \quad \Longrightarrow \quad B=-11.
   $$

Thus, we have found:

$$
Q(x)= 3x-1, \quad A=33, \quad B=-11.
$$
$$ Q(x) = 3x - 1 $$, $$ A = 33 $$, $$ B = -11 $$.

Given , where is a polynomial.
Find
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