Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To find the remainder when dividing the polynomial \(4x^4 - 6x^3 + 6x^2 - 1\) by the polynomial \(2x^2 - 3\), let's go through the process of polynomial division step-by-step.
### Step 1: Set Up the Division
We start with the dividend \(4x^4 - 6x^3 + 6x^2 - 1\) and the divisor \(2x^2 - 3\).
### Step 2: Divide the Leading Terms
First, we divide the leading term of the dividend, \(4x^4\), by the leading term of the divisor, \(2x^2\):
[tex]\[ \frac{4x^4}{2x^2} = 2x^2 \][/tex]
This gives us the first term of the quotient, which is \(2x^2\).
### Step 3: Multiply and Subtract
Next, we multiply the entire divisor by this term of the quotient:
[tex]\[ (2x^2)(2x^2 - 3) = 4x^4 - 6x^2 \][/tex]
Now, subtract this product from the current dividend:
[tex]\[ (4x^4 - 6x^3 + 6x^2 - 1) - (4x^4 - 6x^2) = -6x^3 + 12x^2 - 1 \][/tex]
### Step 4: Repeat the Process
Now, we repeat the process with the new polynomial \(-6x^3 + 12x^2 - 1\).
#### Divide the Leading Terms
[tex]\[ \frac{-6x^3}{2x^2} = -3x \][/tex]
This gives us the next term of the quotient, which is \(-3x\).
#### Multiply and Subtract
[tex]\[ (-3x)(2x^2 - 3) = -6x^3 + 9x \][/tex]
Now, subtract this product from the current polynomial:
[tex]\[ (-6x^3 + 12x^2 - 1) - (-6x^3 + 9x) = 12x^2 - 9x - 1 \][/tex]
### Step 5: Continue the Process
We continue with the polynomial \(12x^2 - 9x - 1\).
#### Divide the Leading Terms
[tex]\[ \frac{12x^2}{2x^2} = 6 \][/tex]
This gives us the next term of the quotient, which is \(6\).
#### Multiply and Subtract
[tex]\[ (6)(2x^2 - 3) = 12x^2 - 18 \][/tex]
Subtract this product from the new polynomial:
[tex]\[ (12x^2 - 9x - 1) - (12x^2 - 18) = -9x + 17 \][/tex]
### Conclusion
Now, we cannot divide further as the degree of the remainder \(-9x + 17\) is less than the degree of the divisor \(2x^2 - 3\). Therefore, the quotient is \(2x^2 - 3x + 6\), and the remainder is \(17 -9 x\).
The final result of the division is:
[tex]\[ \boxed{(2x^2 - 3x + 6, 17 - 9x)} \][/tex]
where [tex]\(2x^2 - 3x + 6\)[/tex] is the quotient and [tex]\(17 - 9x\)[/tex] is the remainder.
### Step 1: Set Up the Division
We start with the dividend \(4x^4 - 6x^3 + 6x^2 - 1\) and the divisor \(2x^2 - 3\).
### Step 2: Divide the Leading Terms
First, we divide the leading term of the dividend, \(4x^4\), by the leading term of the divisor, \(2x^2\):
[tex]\[ \frac{4x^4}{2x^2} = 2x^2 \][/tex]
This gives us the first term of the quotient, which is \(2x^2\).
### Step 3: Multiply and Subtract
Next, we multiply the entire divisor by this term of the quotient:
[tex]\[ (2x^2)(2x^2 - 3) = 4x^4 - 6x^2 \][/tex]
Now, subtract this product from the current dividend:
[tex]\[ (4x^4 - 6x^3 + 6x^2 - 1) - (4x^4 - 6x^2) = -6x^3 + 12x^2 - 1 \][/tex]
### Step 4: Repeat the Process
Now, we repeat the process with the new polynomial \(-6x^3 + 12x^2 - 1\).
#### Divide the Leading Terms
[tex]\[ \frac{-6x^3}{2x^2} = -3x \][/tex]
This gives us the next term of the quotient, which is \(-3x\).
#### Multiply and Subtract
[tex]\[ (-3x)(2x^2 - 3) = -6x^3 + 9x \][/tex]
Now, subtract this product from the current polynomial:
[tex]\[ (-6x^3 + 12x^2 - 1) - (-6x^3 + 9x) = 12x^2 - 9x - 1 \][/tex]
### Step 5: Continue the Process
We continue with the polynomial \(12x^2 - 9x - 1\).
#### Divide the Leading Terms
[tex]\[ \frac{12x^2}{2x^2} = 6 \][/tex]
This gives us the next term of the quotient, which is \(6\).
#### Multiply and Subtract
[tex]\[ (6)(2x^2 - 3) = 12x^2 - 18 \][/tex]
Subtract this product from the new polynomial:
[tex]\[ (12x^2 - 9x - 1) - (12x^2 - 18) = -9x + 17 \][/tex]
### Conclusion
Now, we cannot divide further as the degree of the remainder \(-9x + 17\) is less than the degree of the divisor \(2x^2 - 3\). Therefore, the quotient is \(2x^2 - 3x + 6\), and the remainder is \(17 -9 x\).
The final result of the division is:
[tex]\[ \boxed{(2x^2 - 3x + 6, 17 - 9x)} \][/tex]
where [tex]\(2x^2 - 3x + 6\)[/tex] is the quotient and [tex]\(17 - 9x\)[/tex] is the remainder.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.