To find the quotient [tex]\( Q \)[/tex] and the remainder [tex]\( R \)[/tex] when dividing the polynomial [tex]\( 24x^3 - 14x^2 + 20x + 6 \)[/tex] by the polynomial [tex]\( 4x^2 - 3x + 5 \)[/tex], we will perform polynomial division.
We are given the polynomials:
[tex]\[ P(x) = 24x^3 - 14x^2 + 20x + 6 \][/tex]
[tex]\[ D(x) = 4x^2 - 3x + 5 \][/tex]
1. First Division Step:
[tex]\[
\frac{24x^3}{4x^2} = 6x
\][/tex]
So, the first term in the quotient is [tex]\( 6x \)[/tex].
2. Multiply and Subtract:
[tex]\[
24x^3 - 14x^2 + 20x + 6 - (6x \cdot (4x^2 - 3x + 5)) = 24x^3 - 14x^2 + 20x + 6 - (24x^3 - 18x^2 + 30x)
\][/tex]
[tex]\[
= 24x^3 - 14x^2 + 20x + 6 - 24x^3 + 18x^2 - 30x
\][/tex]
[tex]\[
= ( -14x^2 + 18x^2 ) + ( 20x - 30x ) + 6
\][/tex]
[tex]\[
= 4x^2 - 10x + 6
\][/tex]
3. Second Division Step:
[tex]\[
\frac{4x^2}{4x^2} = 1
\][/tex]
So, the next term is [tex]\( 1 \)[/tex].
4. Multiply and Subtract:
[tex]\[
4x^2 - 10x + 6 - (1 \cdot (4x^2 - 3x + 5)) = 4x^2 - 10x + 6 - (4x^2 - 3x + 5)
\][/tex]
[tex]\[
= 4x^2 - 10x + 6 - 4x^2 + 3x - 5
\][/tex]
[tex]\[
= ( -10x + 3x ) + ( 6 - 5 )
\][/tex]
[tex]\[
= -7x + 1
\][/tex]
So, the quotient [tex]\( Q \)[/tex] is:
[tex]\[ Q = 6x + 1 \][/tex]
And the remainder [tex]\( R \)[/tex] is:
[tex]\[ R = -7x + 1 \][/tex]
Therefore, presenting the result, we have:
[tex]\[ Q = 6x + 1 \][/tex]
[tex]\[ R = -7x + 1 \][/tex]
So, the division [tex]\( (24x^3 - 14x^2 + 20x + 6) \div (4x^2 - 3x + 5) \)[/tex] yields:
[tex]\[ \left(24 x^3 - 14 x^2 + 20 x + 6\right) \div \left(4 x^2 - 3 x + 5\right) = (6x + 1) + \frac{-7x + 1}{4 x^2 - 3 x + 5} \][/tex]
