Answer :
To divide the polynomial [tex]\( \frac{5x^4 - 19x^3 + 14x^2 - 9x + 9}{x - 3} \)[/tex], we can use polynomial long division. Here is a detailed, step-by-step process to get the solution:
1. Set up the division: Write the polynomial [tex]\( 5x^4 - 19x^3 + 14x^2 - 9x + 9 \)[/tex] under the division symbol, and the divisor [tex]\( x - 3 \)[/tex] to the left.
2. Divide the first term: Divide the leading term of the polynomial (which is [tex]\( 5x^4 \)[/tex]) by the leading term of the divisor (which is [tex]\( x \)[/tex]).
[tex]\[
\frac{5x^4}{x} = 5x^3
\][/tex]
So, the first term of the quotient is [tex]\( 5x^3 \)[/tex].
3. Multiply and subtract: Multiply [tex]\( 5x^3 \)[/tex] by the divisor [tex]\( x - 3 \)[/tex] and subtract the result from the original polynomial.
[tex]\[
(5x^3)(x - 3) = 5x^4 - 15x^3
\][/tex]
Subtract:
[tex]\[
(5x^4 - 19x^3) - (5x^4 - 15x^3) = -4x^3
\][/tex]
4. Bring down the next term: Bring down the next term from the polynomial to get [tex]\(-4x^3 + 14x^2\)[/tex].
5. Repeat the process: Divide the new leading term [tex]\(-4x^3\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{-4x^3}{x} = -4x^2
\][/tex]
Multiply and subtract:
[tex]\[
(-4x^2)(x - 3) = -4x^3 + 12x^2
\][/tex]
Subtract:
[tex]\[
(-4x^3 + 14x^2) - (-4x^3 + 12x^2) = 2x^2
\][/tex]
6. Bring down the next term: Bring down the next term to make [tex]\(2x^2 - 9x\)[/tex].
7. Continue dividing: Divide the leading term [tex]\(2x^2\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{2x^2}{x} = 2x
\][/tex]
Multiply and subtract:
[tex]\[
(2x)(x - 3) = 2x^2 - 6x
\][/tex]
Subtract:
[tex]\[
(2x^2 - 9x) - (2x^2 - 6x) = -3x
\][/tex]
8. Bring down the last term: Bring down the final term, giving [tex]\(-3x + 9\)[/tex].
9. Final division step: Divide [tex]\(-3x\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{-3x}{x} = -3
\][/tex]
Multiply and subtract:
[tex]\[
(-3)(x - 3) = -3x + 9
\][/tex]
Subtract:
[tex]\[
(-3x + 9) - (-3x + 9) = 0
\][/tex]
After completing all these steps, we find that the remainder is 0, and the quotient is:
[tex]\[
5x^3 - 4x^2 + 2x - 3
\][/tex]
Therefore, the result of [tex]\( \frac{5x^4 - 19x^3 + 14x^2 - 9x + 9}{x - 3} \)[/tex] is:
[tex]\[
5x^3 - 4x^2 + 2x - 3
\][/tex]
1. Set up the division: Write the polynomial [tex]\( 5x^4 - 19x^3 + 14x^2 - 9x + 9 \)[/tex] under the division symbol, and the divisor [tex]\( x - 3 \)[/tex] to the left.
2. Divide the first term: Divide the leading term of the polynomial (which is [tex]\( 5x^4 \)[/tex]) by the leading term of the divisor (which is [tex]\( x \)[/tex]).
[tex]\[
\frac{5x^4}{x} = 5x^3
\][/tex]
So, the first term of the quotient is [tex]\( 5x^3 \)[/tex].
3. Multiply and subtract: Multiply [tex]\( 5x^3 \)[/tex] by the divisor [tex]\( x - 3 \)[/tex] and subtract the result from the original polynomial.
[tex]\[
(5x^3)(x - 3) = 5x^4 - 15x^3
\][/tex]
Subtract:
[tex]\[
(5x^4 - 19x^3) - (5x^4 - 15x^3) = -4x^3
\][/tex]
4. Bring down the next term: Bring down the next term from the polynomial to get [tex]\(-4x^3 + 14x^2\)[/tex].
5. Repeat the process: Divide the new leading term [tex]\(-4x^3\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{-4x^3}{x} = -4x^2
\][/tex]
Multiply and subtract:
[tex]\[
(-4x^2)(x - 3) = -4x^3 + 12x^2
\][/tex]
Subtract:
[tex]\[
(-4x^3 + 14x^2) - (-4x^3 + 12x^2) = 2x^2
\][/tex]
6. Bring down the next term: Bring down the next term to make [tex]\(2x^2 - 9x\)[/tex].
7. Continue dividing: Divide the leading term [tex]\(2x^2\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{2x^2}{x} = 2x
\][/tex]
Multiply and subtract:
[tex]\[
(2x)(x - 3) = 2x^2 - 6x
\][/tex]
Subtract:
[tex]\[
(2x^2 - 9x) - (2x^2 - 6x) = -3x
\][/tex]
8. Bring down the last term: Bring down the final term, giving [tex]\(-3x + 9\)[/tex].
9. Final division step: Divide [tex]\(-3x\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{-3x}{x} = -3
\][/tex]
Multiply and subtract:
[tex]\[
(-3)(x - 3) = -3x + 9
\][/tex]
Subtract:
[tex]\[
(-3x + 9) - (-3x + 9) = 0
\][/tex]
After completing all these steps, we find that the remainder is 0, and the quotient is:
[tex]\[
5x^3 - 4x^2 + 2x - 3
\][/tex]
Therefore, the result of [tex]\( \frac{5x^4 - 19x^3 + 14x^2 - 9x + 9}{x - 3} \)[/tex] is:
[tex]\[
5x^3 - 4x^2 + 2x - 3
\][/tex]
