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Answer :
To find the result when the polynomial [tex]\(4x^3 - 19x^2 + 20x + 3\)[/tex] is divided by [tex]\(x - 3\)[/tex], we perform polynomial division. Here’s a step-by-step explanation of how this can be done:
1. Set Up the Division: Write the polynomial [tex]\(4x^3 - 19x^2 + 20x + 3\)[/tex] as the dividend inside the division box and [tex]\(x - 3\)[/tex] as the divisor outside.
2. Divide the Leading Terms: Divide the leading term of the dividend, [tex]\(4x^3\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(4x^2\)[/tex]. Write [tex]\(4x^2\)[/tex] as the first term of the quotient.
3. Multiply and Subtract: Multiply the entire divisor [tex]\(x - 3\)[/tex] by the term just found in the quotient, [tex]\(4x^2\)[/tex]. This gives [tex]\(4x^3 - 12x^2\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(4x^3 - 19x^2 + 20x + 3) - (4x^3 - 12x^2) = -7x^2 + 20x + 3
\][/tex]
4. Repeat the Process:
- Divide the new leading term [tex]\(-7x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(-7x\)[/tex].
- Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(-7x\)[/tex], resulting in [tex]\(-7x^2 + 21x\)[/tex].
- Subtract this from the result of the previous subtraction:
[tex]\[
(-7x^2 + 20x + 3) - (-7x^2 + 21x) = -x + 3
\][/tex]
5. Continue:
- Divide the new leading term [tex]\(-x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(-1\)[/tex].
- Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(-1\)[/tex], resulting in [tex]\(-x + 3\)[/tex].
- Subtract this from the result of the previous subtraction:
[tex]\[
(-x + 3) - (-x + 3) = 0
\][/tex]
Since our remainder is 0, the division is complete, and the quotient is [tex]\(4x^2 - 7x - 1\)[/tex].
Therefore, the result of dividing [tex]\(4x^3 - 19x^2 + 20x + 3\)[/tex] by [tex]\(x - 3\)[/tex] is the quotient:
[tex]\[
4x^2 - 7x - 1
\][/tex]
And because the remainder is 0, the divisor [tex]\(x-3\)[/tex] perfectly divides the polynomial.
1. Set Up the Division: Write the polynomial [tex]\(4x^3 - 19x^2 + 20x + 3\)[/tex] as the dividend inside the division box and [tex]\(x - 3\)[/tex] as the divisor outside.
2. Divide the Leading Terms: Divide the leading term of the dividend, [tex]\(4x^3\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(4x^2\)[/tex]. Write [tex]\(4x^2\)[/tex] as the first term of the quotient.
3. Multiply and Subtract: Multiply the entire divisor [tex]\(x - 3\)[/tex] by the term just found in the quotient, [tex]\(4x^2\)[/tex]. This gives [tex]\(4x^3 - 12x^2\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(4x^3 - 19x^2 + 20x + 3) - (4x^3 - 12x^2) = -7x^2 + 20x + 3
\][/tex]
4. Repeat the Process:
- Divide the new leading term [tex]\(-7x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(-7x\)[/tex].
- Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(-7x\)[/tex], resulting in [tex]\(-7x^2 + 21x\)[/tex].
- Subtract this from the result of the previous subtraction:
[tex]\[
(-7x^2 + 20x + 3) - (-7x^2 + 21x) = -x + 3
\][/tex]
5. Continue:
- Divide the new leading term [tex]\(-x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(-1\)[/tex].
- Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(-1\)[/tex], resulting in [tex]\(-x + 3\)[/tex].
- Subtract this from the result of the previous subtraction:
[tex]\[
(-x + 3) - (-x + 3) = 0
\][/tex]
Since our remainder is 0, the division is complete, and the quotient is [tex]\(4x^2 - 7x - 1\)[/tex].
Therefore, the result of dividing [tex]\(4x^3 - 19x^2 + 20x + 3\)[/tex] by [tex]\(x - 3\)[/tex] is the quotient:
[tex]\[
4x^2 - 7x - 1
\][/tex]
And because the remainder is 0, the divisor [tex]\(x-3\)[/tex] perfectly divides the polynomial.
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