Answer :
To solve the division of the polynomial [tex]\((x^2 + 5x - 14)\)[/tex] by [tex]\((x - 2)\)[/tex], we can use polynomial long division. Here's a step-by-step explanation:
1. Set up the division: Divide the polynomial [tex]\(x^2 + 5x - 14\)[/tex] by [tex]\(x - 2\)[/tex].
2. Divide the first term: Divide the first term of the dividend ([tex]\(x^2\)[/tex]) by the first term of the divisor ([tex]\(x\)[/tex]). This gives you [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x - 2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(x^2 - 2x\)[/tex]. Subtract this result from [tex]\(x^2 + 5x - 14\)[/tex]:
[tex]\[
(x^2 + 5x - 14) - (x^2 - 2x) = 7x - 14
\][/tex]
4. Repeat for the next term: Now, divide the first term of the new polynomial [tex]\(7x\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]. This gives you [tex]\(+7\)[/tex].
5. Multiply and subtract again: Multiply the divisor [tex]\(x - 2\)[/tex] by [tex]\(7\)[/tex], resulting in [tex]\(7x - 14\)[/tex]. Subtract this from [tex]\(7x - 14\)[/tex]:
[tex]\[
(7x - 14) - (7x - 14) = 0
\][/tex]
There are no remaining terms; therefore, the remainder is 0.
6. Conclusion: The quotient of the division is [tex]\(x + 7\)[/tex], and since the remainder is 0, [tex]\(x + 7\)[/tex] is the exact result of dividing [tex]\(x^2 + 5x - 14\)[/tex] by [tex]\(x - 2\)[/tex].
So, the solution to the division is [tex]\(x + 7\)[/tex].
1. Set up the division: Divide the polynomial [tex]\(x^2 + 5x - 14\)[/tex] by [tex]\(x - 2\)[/tex].
2. Divide the first term: Divide the first term of the dividend ([tex]\(x^2\)[/tex]) by the first term of the divisor ([tex]\(x\)[/tex]). This gives you [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x - 2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(x^2 - 2x\)[/tex]. Subtract this result from [tex]\(x^2 + 5x - 14\)[/tex]:
[tex]\[
(x^2 + 5x - 14) - (x^2 - 2x) = 7x - 14
\][/tex]
4. Repeat for the next term: Now, divide the first term of the new polynomial [tex]\(7x\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]. This gives you [tex]\(+7\)[/tex].
5. Multiply and subtract again: Multiply the divisor [tex]\(x - 2\)[/tex] by [tex]\(7\)[/tex], resulting in [tex]\(7x - 14\)[/tex]. Subtract this from [tex]\(7x - 14\)[/tex]:
[tex]\[
(7x - 14) - (7x - 14) = 0
\][/tex]
There are no remaining terms; therefore, the remainder is 0.
6. Conclusion: The quotient of the division is [tex]\(x + 7\)[/tex], and since the remainder is 0, [tex]\(x + 7\)[/tex] is the exact result of dividing [tex]\(x^2 + 5x - 14\)[/tex] by [tex]\(x - 2\)[/tex].
So, the solution to the division is [tex]\(x + 7\)[/tex].
