Answer :
To rewrite the given expression [tex]\(\frac{10x^4 - 23x^3 + 18x^2 - 15x + 5}{5x^2 - 4x + 3}\)[/tex] in the form [tex]\(q(x) + \frac{r(x)}{5x^2 - 4x + 3}\)[/tex], we need to perform polynomial division. Here’s how we can break it down step-by-step:
1. Identify the Divisor and Dividend
The dividend (the polynomial being divided) is [tex]\(10x^4 - 23x^3 + 18x^2 - 15x + 5\)[/tex].
The divisor is [tex]\(5x^2 - 4x + 3\)[/tex].
2. Divide the Leading Terms
Begin by dividing the leading term of the dividend, [tex]\(10x^4\)[/tex], by the leading term of the divisor, [tex]\(5x^2\)[/tex]. This gives [tex]\(2x^2\)[/tex].
3. Multiply and Subtract
Multiply [tex]\(2x^2\)[/tex] by the entire divisor:
[tex]\((2x^2)(5x^2 - 4x + 3) = 10x^4 - 8x^3 + 6x^2\)[/tex].
Subtract this from the original dividend:
[tex]\((10x^4 - 23x^3 + 18x^2 - 15x + 5) - (10x^4 - 8x^3 + 6x^2)\)[/tex]
Which results in: [tex]\(-15x^3 + 12x^2 - 15x + 5\)[/tex].
4. Repeat the Process
Now divide [tex]\(-15x^3\)[/tex] by [tex]\(5x^2\)[/tex] to get [tex]\(-3x\)[/tex].
5. Multiply and Subtract Again
Multiply [tex]\(-3x\)[/tex] by the divisor:
[tex]\((-3x)(5x^2 - 4x + 3) = -15x^3 + 12x^2 - 9x\)[/tex].
Subtract:
[tex]\((-15x^3 + 12x^2 - 15x + 5) - (-15x^3 + 12x^2 - 9x)\)[/tex]
This gives us [tex]\(-6x + 5\)[/tex].
6. Quotient and Remainder
The quotient from these operations is [tex]\(2x^2 - 3x\)[/tex], and the remainder is [tex]\(-6x + 5\)[/tex].
So, the expression can be rewritten as:
[tex]\[
2x^2 - 3x + \frac{-6x + 5}{5x^2 - 4x + 3}
\][/tex]
This form represents the division of the polynomial where [tex]\(q(x) = 2x^2 - 3x\)[/tex] is the quotient and [tex]\(r(x) = -6x + 5\)[/tex] is the remainder.
1. Identify the Divisor and Dividend
The dividend (the polynomial being divided) is [tex]\(10x^4 - 23x^3 + 18x^2 - 15x + 5\)[/tex].
The divisor is [tex]\(5x^2 - 4x + 3\)[/tex].
2. Divide the Leading Terms
Begin by dividing the leading term of the dividend, [tex]\(10x^4\)[/tex], by the leading term of the divisor, [tex]\(5x^2\)[/tex]. This gives [tex]\(2x^2\)[/tex].
3. Multiply and Subtract
Multiply [tex]\(2x^2\)[/tex] by the entire divisor:
[tex]\((2x^2)(5x^2 - 4x + 3) = 10x^4 - 8x^3 + 6x^2\)[/tex].
Subtract this from the original dividend:
[tex]\((10x^4 - 23x^3 + 18x^2 - 15x + 5) - (10x^4 - 8x^3 + 6x^2)\)[/tex]
Which results in: [tex]\(-15x^3 + 12x^2 - 15x + 5\)[/tex].
4. Repeat the Process
Now divide [tex]\(-15x^3\)[/tex] by [tex]\(5x^2\)[/tex] to get [tex]\(-3x\)[/tex].
5. Multiply and Subtract Again
Multiply [tex]\(-3x\)[/tex] by the divisor:
[tex]\((-3x)(5x^2 - 4x + 3) = -15x^3 + 12x^2 - 9x\)[/tex].
Subtract:
[tex]\((-15x^3 + 12x^2 - 15x + 5) - (-15x^3 + 12x^2 - 9x)\)[/tex]
This gives us [tex]\(-6x + 5\)[/tex].
6. Quotient and Remainder
The quotient from these operations is [tex]\(2x^2 - 3x\)[/tex], and the remainder is [tex]\(-6x + 5\)[/tex].
So, the expression can be rewritten as:
[tex]\[
2x^2 - 3x + \frac{-6x + 5}{5x^2 - 4x + 3}
\][/tex]
This form represents the division of the polynomial where [tex]\(q(x) = 2x^2 - 3x\)[/tex] is the quotient and [tex]\(r(x) = -6x + 5\)[/tex] is the remainder.
