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Answer :
To determine whether the binomial [tex]\(x+3\)[/tex] is a factor of the polynomial [tex]\(x^3+9x^2+10x-25\)[/tex], we can use the concept of polynomial division and the Remainder Theorem.
Here’s a step-by-step explanation of how to do it:
1. Set up the division: We want to divide the polynomial [tex]\(x^3 + 9x^2 + 10x - 25\)[/tex] by the binomial [tex]\(x + 3\)[/tex].
2. Perform polynomial division:
- Divide the first term: Look at the leading term of the polynomial, which is [tex]\(x^3\)[/tex]. Divide it by the leading term of the divisor, [tex]\(x\)[/tex], to get [tex]\(x^2\)[/tex].
- Multiply and subtract: Multiply [tex]\(x^2\)[/tex] by [tex]\(x + 3\)[/tex] to get [tex]\(x^3 + 3x^2\)[/tex]. Subtract this from the original polynomial to eliminate the [tex]\(x^3\)[/tex] term, leaving [tex]\(6x^2 + 10x - 25\)[/tex].
- Repeat the process:
- Divide [tex]\(6x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(6x\)[/tex].
- Multiply [tex]\(6x\)[/tex] by [tex]\(x + 3\)[/tex] to get [tex]\(6x^2 + 18x\)[/tex]. Subtract this from the result to leave [tex]\(-8x - 25\)[/tex].
- Finally, divide [tex]\(-8x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-8\)[/tex].
- Multiply [tex]\(-8\)[/tex] by [tex]\(x + 3\)[/tex] to get [tex]\(-8x - 24\)[/tex]. Subtracting gives a remainder of [tex]\(-1\)[/tex].
3. Check the remainder: After performing the division, the remainder is [tex]\(-1\)[/tex].
4. Conclusion using the Remainder Theorem: According to the Remainder Theorem, if you divide a polynomial [tex]\(f(x)\)[/tex] by [tex]\(x - c\)[/tex] and get a remainder of 0, then [tex]\(x - c\)[/tex] is a factor of the polynomial. Since we didn't get a remainder of 0 (we got [tex]\(-1\)[/tex]), [tex]\(x + 3\)[/tex] is not a factor of the polynomial [tex]\(x^3 + 9x^2 + 10x - 25\)[/tex].
Therefore, the answer is No, [tex]\(x+3\)[/tex] is not a factor of [tex]\(x^3 + 9x^2 + 10x - 25\)[/tex].
Here’s a step-by-step explanation of how to do it:
1. Set up the division: We want to divide the polynomial [tex]\(x^3 + 9x^2 + 10x - 25\)[/tex] by the binomial [tex]\(x + 3\)[/tex].
2. Perform polynomial division:
- Divide the first term: Look at the leading term of the polynomial, which is [tex]\(x^3\)[/tex]. Divide it by the leading term of the divisor, [tex]\(x\)[/tex], to get [tex]\(x^2\)[/tex].
- Multiply and subtract: Multiply [tex]\(x^2\)[/tex] by [tex]\(x + 3\)[/tex] to get [tex]\(x^3 + 3x^2\)[/tex]. Subtract this from the original polynomial to eliminate the [tex]\(x^3\)[/tex] term, leaving [tex]\(6x^2 + 10x - 25\)[/tex].
- Repeat the process:
- Divide [tex]\(6x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(6x\)[/tex].
- Multiply [tex]\(6x\)[/tex] by [tex]\(x + 3\)[/tex] to get [tex]\(6x^2 + 18x\)[/tex]. Subtract this from the result to leave [tex]\(-8x - 25\)[/tex].
- Finally, divide [tex]\(-8x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-8\)[/tex].
- Multiply [tex]\(-8\)[/tex] by [tex]\(x + 3\)[/tex] to get [tex]\(-8x - 24\)[/tex]. Subtracting gives a remainder of [tex]\(-1\)[/tex].
3. Check the remainder: After performing the division, the remainder is [tex]\(-1\)[/tex].
4. Conclusion using the Remainder Theorem: According to the Remainder Theorem, if you divide a polynomial [tex]\(f(x)\)[/tex] by [tex]\(x - c\)[/tex] and get a remainder of 0, then [tex]\(x - c\)[/tex] is a factor of the polynomial. Since we didn't get a remainder of 0 (we got [tex]\(-1\)[/tex]), [tex]\(x + 3\)[/tex] is not a factor of the polynomial [tex]\(x^3 + 9x^2 + 10x - 25\)[/tex].
Therefore, the answer is No, [tex]\(x+3\)[/tex] is not a factor of [tex]\(x^3 + 9x^2 + 10x - 25\)[/tex].
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