Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To divide the polynomial [tex]\(x^2 + 2x + 6\)[/tex] by [tex]\(x - 3\)[/tex], we will perform polynomial long division step-by-step.
1. Set up the division:
[tex]\[ \begin{array}{r|l} x - 3 & x^2 + 2x + 6 \\ \end{array} \][/tex]
2. Divide the first term of the numerator by the first term of the divisor:
- Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{x^2}{x} = x \][/tex]
- Write [tex]\(x\)[/tex] above the division line:
[tex]\[ \begin{array}{r|ll} x - 3 & x^2 + 2x + 6 \\ & x \\ \end{array} \][/tex]
3. Multiply the entire divisor by this result:
[tex]\[ x \cdot (x - 3) = x^2 - 3x \][/tex]
4. Subtract this product from the original polynomial:
[tex]\[ \begin{array}{r|ll} x - 3 & x^2 + 2x + 6 \\ & x^2 - 3x \\ \hline & 5x + 6 \\ \end{array} \][/tex]
5. Bring down the next term (if any).
- Move down the [tex]\(5x + 6\)[/tex] completely.
6. Repeat the division with the new polynomial:
- Divide [tex]\(5x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{5x}{x} = 5 \][/tex]
- Write [tex]\(5\)[/tex] above the division line, next to [tex]\(x\)[/tex]:
[tex]\[ \begin{array}{r|lll} x - 3 & x^2 + 2x + 6 \\ & x + 5 \\ \end{array} \][/tex]
7. Multiply the entire divisor by this result:
[tex]\[ 5 \cdot (x - 3) = 5x - 15 \][/tex]
8. Subtract this product from the new polynomial:
[tex]\[ \begin{array}{r|lll} x - 3 & x^2 + 2x + 6 \\ & x^2 - 3x \\ \hline & 5x + 6 \\ & 5x - 15 \\ \hline & 21 \\ \end{array} \][/tex]
Rewriting the remainders and quotients, we get:
[tex]\[ x^2 + 2x + 6 = (x - 3)(x + 5) + 21 \][/tex]
So, the quotient is [tex]\(x + 5\)[/tex] and the remainder is [tex]\(21\)[/tex].
Therefore, [tex]\((x^2 + 2x + 6) \div (x - 3) = x + 5\)[/tex] with a remainder of [tex]\(21\)[/tex].
1. Set up the division:
[tex]\[ \begin{array}{r|l} x - 3 & x^2 + 2x + 6 \\ \end{array} \][/tex]
2. Divide the first term of the numerator by the first term of the divisor:
- Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{x^2}{x} = x \][/tex]
- Write [tex]\(x\)[/tex] above the division line:
[tex]\[ \begin{array}{r|ll} x - 3 & x^2 + 2x + 6 \\ & x \\ \end{array} \][/tex]
3. Multiply the entire divisor by this result:
[tex]\[ x \cdot (x - 3) = x^2 - 3x \][/tex]
4. Subtract this product from the original polynomial:
[tex]\[ \begin{array}{r|ll} x - 3 & x^2 + 2x + 6 \\ & x^2 - 3x \\ \hline & 5x + 6 \\ \end{array} \][/tex]
5. Bring down the next term (if any).
- Move down the [tex]\(5x + 6\)[/tex] completely.
6. Repeat the division with the new polynomial:
- Divide [tex]\(5x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{5x}{x} = 5 \][/tex]
- Write [tex]\(5\)[/tex] above the division line, next to [tex]\(x\)[/tex]:
[tex]\[ \begin{array}{r|lll} x - 3 & x^2 + 2x + 6 \\ & x + 5 \\ \end{array} \][/tex]
7. Multiply the entire divisor by this result:
[tex]\[ 5 \cdot (x - 3) = 5x - 15 \][/tex]
8. Subtract this product from the new polynomial:
[tex]\[ \begin{array}{r|lll} x - 3 & x^2 + 2x + 6 \\ & x^2 - 3x \\ \hline & 5x + 6 \\ & 5x - 15 \\ \hline & 21 \\ \end{array} \][/tex]
Rewriting the remainders and quotients, we get:
[tex]\[ x^2 + 2x + 6 = (x - 3)(x + 5) + 21 \][/tex]
So, the quotient is [tex]\(x + 5\)[/tex] and the remainder is [tex]\(21\)[/tex].
Therefore, [tex]\((x^2 + 2x + 6) \div (x - 3) = x + 5\)[/tex] with a remainder of [tex]\(21\)[/tex].
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.