We appreciate your visit to What is the remainder when tex 3x 3 2x 2 4x 3 tex is divided by tex x 2 3x 3 tex A 30 B. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To find the remainder when
[tex]$$
3x^3 - 2x^2 + 4x - 3
$$[/tex]
is divided by
[tex]$$
x^2 + 3x + 3,
$$[/tex]
we can perform polynomial long division. Here’s a detailed step-by-step explanation:
1. Set up the division:
Divide the polynomial
[tex]$$
f(x) = 3x^3 - 2x^2 + 4x - 3
$$[/tex]
by the divisor
[tex]$$
g(x) = x^2 + 3x + 3.
$$[/tex]
2. Divide the leading terms:
The first term in [tex]$f(x)$[/tex] is [tex]$3x^3$[/tex] and the first term in [tex]$g(x)$[/tex] is [tex]$x^2$[/tex]. Dividing these gives
[tex]$$
\frac{3x^3}{x^2} = 3x.
$$[/tex]
This [tex]$3x$[/tex] is the first term of the quotient.
3. Multiply and subtract:
Multiply the entire divisor by [tex]$3x$[/tex]:
[tex]$$
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x.
$$[/tex]
Subtract this from [tex]$f(x)$[/tex]:
[tex]\[
\begin{aligned}
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) &= (3x^3 - 3x^3) + (-2x^2 - 9x^2) + (4x - 9x) - 3 \\
&= -11x^2 - 5x - 3.
\end{aligned}
\][/tex]
4. Proceed with the next division:
The new polynomial to divide is
[tex]$$
-11x^2 - 5x - 3.
$$[/tex]
Divide the leading term [tex]$-11x^2$[/tex] by the leading term [tex]$x^2$[/tex] of the divisor:
[tex]$$
\frac{-11x^2}{x^2} = -11.
$$[/tex]
This [tex]$-11$[/tex] is the next term of the quotient.
5. Multiply and subtract again:
Multiply the divisor by [tex]$-11$[/tex]:
[tex]$$
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33.
$$[/tex]
Subtract this product from the current remainder:
[tex]\[
\begin{aligned}
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) &= (-11x^2 + 11x^2) + (-5x + 33x) + (-3 + 33) \\
&= 28x + 30.
\end{aligned}
\][/tex]
6. Identify the remainder:
Since the divisor has degree 2 and the resulting expression [tex]$28x + 30$[/tex] is of degree 1 (which is less than 2), we have reached the remainder.
Thus, the division yields a quotient of
[tex]$$
3x - 11
$$[/tex]
and a remainder of
[tex]$$
28x + 30.
$$[/tex]
Therefore, the remainder when [tex]$3x^3 - 2x^2 + 4x - 3$[/tex] is divided by [tex]$x^2 + 3x + 3$[/tex] is
[tex]$$
\boxed{28x + 30}.
$$[/tex]
[tex]$$
3x^3 - 2x^2 + 4x - 3
$$[/tex]
is divided by
[tex]$$
x^2 + 3x + 3,
$$[/tex]
we can perform polynomial long division. Here’s a detailed step-by-step explanation:
1. Set up the division:
Divide the polynomial
[tex]$$
f(x) = 3x^3 - 2x^2 + 4x - 3
$$[/tex]
by the divisor
[tex]$$
g(x) = x^2 + 3x + 3.
$$[/tex]
2. Divide the leading terms:
The first term in [tex]$f(x)$[/tex] is [tex]$3x^3$[/tex] and the first term in [tex]$g(x)$[/tex] is [tex]$x^2$[/tex]. Dividing these gives
[tex]$$
\frac{3x^3}{x^2} = 3x.
$$[/tex]
This [tex]$3x$[/tex] is the first term of the quotient.
3. Multiply and subtract:
Multiply the entire divisor by [tex]$3x$[/tex]:
[tex]$$
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x.
$$[/tex]
Subtract this from [tex]$f(x)$[/tex]:
[tex]\[
\begin{aligned}
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) &= (3x^3 - 3x^3) + (-2x^2 - 9x^2) + (4x - 9x) - 3 \\
&= -11x^2 - 5x - 3.
\end{aligned}
\][/tex]
4. Proceed with the next division:
The new polynomial to divide is
[tex]$$
-11x^2 - 5x - 3.
$$[/tex]
Divide the leading term [tex]$-11x^2$[/tex] by the leading term [tex]$x^2$[/tex] of the divisor:
[tex]$$
\frac{-11x^2}{x^2} = -11.
$$[/tex]
This [tex]$-11$[/tex] is the next term of the quotient.
5. Multiply and subtract again:
Multiply the divisor by [tex]$-11$[/tex]:
[tex]$$
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33.
$$[/tex]
Subtract this product from the current remainder:
[tex]\[
\begin{aligned}
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) &= (-11x^2 + 11x^2) + (-5x + 33x) + (-3 + 33) \\
&= 28x + 30.
\end{aligned}
\][/tex]
6. Identify the remainder:
Since the divisor has degree 2 and the resulting expression [tex]$28x + 30$[/tex] is of degree 1 (which is less than 2), we have reached the remainder.
Thus, the division yields a quotient of
[tex]$$
3x - 11
$$[/tex]
and a remainder of
[tex]$$
28x + 30.
$$[/tex]
Therefore, the remainder when [tex]$3x^3 - 2x^2 + 4x - 3$[/tex] is divided by [tex]$x^2 + 3x + 3$[/tex] is
[tex]$$
\boxed{28x + 30}.
$$[/tex]
Thanks for taking the time to read What is the remainder when tex 3x 3 2x 2 4x 3 tex is divided by tex x 2 3x 3 tex A 30 B. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
