Answer :
To solve the polynomial division [tex]\((5x^3 - 25x^2 + 38x - 26) \div (x - 3)\)[/tex], we can use long division for polynomials. Here is a step-by-step explanation:
1. Set up the division: Write the dividend [tex]\(5x^3 - 25x^2 + 38x - 26\)[/tex] and the divisor [tex]\(x - 3\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives you [tex]\(5x^2\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(5x^2\)[/tex] by the divisor [tex]\(x - 3\)[/tex] to get [tex]\(5x^3 - 15x^2\)[/tex].
- Subtract this from the original polynomial to get a new polynomial: [tex]\(-25x^2 - (5x^3 - 15x^2) = -10x^2\)[/tex].
4. Bring down the next term: Bring down the next term from the original polynomial, which is [tex]\(+38x\)[/tex], to combine it with the [tex]\(-10x^2\)[/tex], resulting in [tex]\(-10x^2 + 38x\)[/tex].
5. Repeat the process:
- Divide the leading term [tex]\(-10x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], giving [tex]\(-10x\)[/tex].
- Multiply [tex]\(-10x\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(-10x^2 + 30x\)[/tex].
- Subtract this from the current polynomial: [tex]\(38x - ( -10x^2 + 30x) = 8x\)[/tex].
6. Bring down the last term: Bring down [tex]\(-26\)[/tex] to the current expression [tex]\(8x\)[/tex], resulting in [tex]\(8x - 26\)[/tex].
7. Final division:
- Divide the leading term [tex]\(8x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], resulting in [tex]\(8\)[/tex].
- Multiply [tex]\(8\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(8x - 24\)[/tex].
- Subtract this from [tex]\(8x - 26\)[/tex] to get a remainder of [tex]\(-2\)[/tex].
Hence, after performing the division, the quotient is [tex]\(5x^2 - 10x + 8\)[/tex], and the remainder is [tex]\(-2\)[/tex].
So, the result of dividing [tex]\((5x^3 - 25x^2 + 38x - 26)\)[/tex] by [tex]\((x - 3)\)[/tex] is a quotient of [tex]\(5x^2 - 10x + 8\)[/tex] and a remainder of [tex]\(-2\)[/tex].
1. Set up the division: Write the dividend [tex]\(5x^3 - 25x^2 + 38x - 26\)[/tex] and the divisor [tex]\(x - 3\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives you [tex]\(5x^2\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(5x^2\)[/tex] by the divisor [tex]\(x - 3\)[/tex] to get [tex]\(5x^3 - 15x^2\)[/tex].
- Subtract this from the original polynomial to get a new polynomial: [tex]\(-25x^2 - (5x^3 - 15x^2) = -10x^2\)[/tex].
4. Bring down the next term: Bring down the next term from the original polynomial, which is [tex]\(+38x\)[/tex], to combine it with the [tex]\(-10x^2\)[/tex], resulting in [tex]\(-10x^2 + 38x\)[/tex].
5. Repeat the process:
- Divide the leading term [tex]\(-10x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], giving [tex]\(-10x\)[/tex].
- Multiply [tex]\(-10x\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(-10x^2 + 30x\)[/tex].
- Subtract this from the current polynomial: [tex]\(38x - ( -10x^2 + 30x) = 8x\)[/tex].
6. Bring down the last term: Bring down [tex]\(-26\)[/tex] to the current expression [tex]\(8x\)[/tex], resulting in [tex]\(8x - 26\)[/tex].
7. Final division:
- Divide the leading term [tex]\(8x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], resulting in [tex]\(8\)[/tex].
- Multiply [tex]\(8\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(8x - 24\)[/tex].
- Subtract this from [tex]\(8x - 26\)[/tex] to get a remainder of [tex]\(-2\)[/tex].
Hence, after performing the division, the quotient is [tex]\(5x^2 - 10x + 8\)[/tex], and the remainder is [tex]\(-2\)[/tex].
So, the result of dividing [tex]\((5x^3 - 25x^2 + 38x - 26)\)[/tex] by [tex]\((x - 3)\)[/tex] is a quotient of [tex]\(5x^2 - 10x + 8\)[/tex] and a remainder of [tex]\(-2\)[/tex].
