Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To find the remainder when dividing [tex]\( 5x^3 + 7x + 5 \)[/tex] by [tex]\( x + 2 \)[/tex], we will perform polynomial division, a method similar to long division for numbers. Here is a step-by-step solution:
### Step-by-Step Solution
1. Set Up the Polynomial Division:
The dividend (the polynomial we are dividing) is [tex]\( 5x^3 + 7x + 5 \)[/tex].
The divisor (the polynomial we are dividing by) is [tex]\( x + 2 \)[/tex].
2. Perform the Division:
Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{5x^3}{x} = 5x^2 \][/tex]
So, the first term of the quotient is [tex]\( 5x^2 \)[/tex].
Step 2: Multiply the entire divisor [tex]\( x + 2 \)[/tex] by this term [tex]\( 5x^2 \)[/tex]:
[tex]\[ (x + 2) \cdot 5x^2 = 5x^3 + 10x^2 \][/tex]
Step 3: Subtract this result from the original dividend:
[tex]\[ (5x^3 + 0x^2 + 7x + 5) - (5x^3 + 10x^2) = -10x^2 + 7x + 5 \][/tex]
Step 4: Repeat the process with the new polynomial [tex]\(-10x^2 + 7x + 5\)[/tex].
Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[ \frac{-10x^2}{x} = -10x \][/tex]
So, the next term of the quotient is [tex]\(-10x\)[/tex].
Step 5: Multiply the divisor by this new term [tex]\(-10x\)[/tex]:
[tex]\[ (x + 2) \cdot (-10x) = -10x^2 - 20x \][/tex]
Step 6: Subtract this result from the new dividend:
[tex]\[ (-10x^2 + 7x + 5) - (-10x^2 - 20x) = 27x + 5 \][/tex]
Step 7: Repeat the process with the new polynomial [tex]\(27x + 5\)[/tex].
Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[ \frac{27x}{x} = 27 \][/tex]
So, the next term of the quotient is [tex]\(27\)[/tex].
Step 8: Multiply the divisor by this new term [tex]\(27\)[/tex]:
[tex]\[ (x + 2) \cdot 27 = 27x + 54 \][/tex]
Step 9: Subtract this result from the new dividend:
[tex]\[ (27x + 5) - (27x + 54) = 5 - 54 = -49 \][/tex]
3. Result:
At this point, no further division is possible because the degree of the new polynomial is less than the degree of the divisor [tex]\( x + 2 \)[/tex]. Thus, the remainder is:
[tex]\[ -49 \][/tex]
### Final Answer
The remainder of the division of [tex]\( \frac{5x^3 + 7x + 5}{x + 2} \)[/tex] is [tex]\( -49 \)[/tex].
### Step-by-Step Solution
1. Set Up the Polynomial Division:
The dividend (the polynomial we are dividing) is [tex]\( 5x^3 + 7x + 5 \)[/tex].
The divisor (the polynomial we are dividing by) is [tex]\( x + 2 \)[/tex].
2. Perform the Division:
Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{5x^3}{x} = 5x^2 \][/tex]
So, the first term of the quotient is [tex]\( 5x^2 \)[/tex].
Step 2: Multiply the entire divisor [tex]\( x + 2 \)[/tex] by this term [tex]\( 5x^2 \)[/tex]:
[tex]\[ (x + 2) \cdot 5x^2 = 5x^3 + 10x^2 \][/tex]
Step 3: Subtract this result from the original dividend:
[tex]\[ (5x^3 + 0x^2 + 7x + 5) - (5x^3 + 10x^2) = -10x^2 + 7x + 5 \][/tex]
Step 4: Repeat the process with the new polynomial [tex]\(-10x^2 + 7x + 5\)[/tex].
Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[ \frac{-10x^2}{x} = -10x \][/tex]
So, the next term of the quotient is [tex]\(-10x\)[/tex].
Step 5: Multiply the divisor by this new term [tex]\(-10x\)[/tex]:
[tex]\[ (x + 2) \cdot (-10x) = -10x^2 - 20x \][/tex]
Step 6: Subtract this result from the new dividend:
[tex]\[ (-10x^2 + 7x + 5) - (-10x^2 - 20x) = 27x + 5 \][/tex]
Step 7: Repeat the process with the new polynomial [tex]\(27x + 5\)[/tex].
Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[ \frac{27x}{x} = 27 \][/tex]
So, the next term of the quotient is [tex]\(27\)[/tex].
Step 8: Multiply the divisor by this new term [tex]\(27\)[/tex]:
[tex]\[ (x + 2) \cdot 27 = 27x + 54 \][/tex]
Step 9: Subtract this result from the new dividend:
[tex]\[ (27x + 5) - (27x + 54) = 5 - 54 = -49 \][/tex]
3. Result:
At this point, no further division is possible because the degree of the new polynomial is less than the degree of the divisor [tex]\( x + 2 \)[/tex]. Thus, the remainder is:
[tex]\[ -49 \][/tex]
### Final Answer
The remainder of the division of [tex]\( \frac{5x^3 + 7x + 5}{x + 2} \)[/tex] is [tex]\( -49 \)[/tex].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.