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Sagot :
Sure, let's solve this step-by-step.
### Step 1: Check if the divisor is in [tex]\( (x - r) \)[/tex] form
For a divisor to be in the form [tex]\( (x - r) \)[/tex], it should look something like [tex]\( x - r \)[/tex] where [tex]\( r \)[/tex] is a constant. In this case, the divisor is [tex]\( -\frac{1}{4}x + 6 \)[/tex].
Here the term with [tex]\( x \)[/tex] is [tex]\(-\frac{1}{4}x\)[/tex], not in the form [tex]\( x-r \)[/tex]. Thus, the divisor is not in the form [tex]\( (x - r) \)[/tex].
Answer: The divisor given is not in [tex]\( (x-r) \)[/tex] form. [tex]\( \square \)[/tex]
### Step 2: Identify the number of terms in the dividend
The dividend polynomial is [tex]\( -x^3 + \frac{101}{4}x^2 - \frac{125}{4}x + 30 \)[/tex].
Count the terms:
1. [tex]\( -x^3 \)[/tex]
2. [tex]\( \frac{101}{4}x^2 \)[/tex]
3. [tex]\( -\frac{125}{4}x \)[/tex]
4. [tex]\( 30 \)[/tex]
So, the dividend has 4 terms.
Answer: There are 4 terms in the dividend. [tex]\( \square \)[/tex]
### Step 3: Perform synthetic division
Given:
- Dividend: [tex]\( -x^3 + \frac{101}{4}x^2 - \frac{125}{4}x + 30 \)[/tex]
- Divisor: [tex]\( -\frac{1}{4}x + 6 \)[/tex]
Here are the coefficients of the dividend: [tex]\(-1, \frac{101}{4}, -\frac{125}{4}, 30\)[/tex].
And the constant term (leading coefficient of [tex]\( x \)[/tex] of the divisor) is [tex]\(-\frac{1}{4}\)[/tex].
#### Synthetic Division Process:
1. Write down the coefficients of the dividend: [tex]\([-1, \frac{101}{4}, -\frac{125}{4}, 30]\)[/tex].
2. The divisor coefficient is [tex]\(-\frac{1}{4}\)[/tex].
Now, start the synthetic division process:
- Initial remainder (first coefficient): -1
#### Process each coefficient step-by-step:
3. Multiply the initial remainder by divisor coefficient and add the next coefficient:
-quotient term: -1
[tex]\[ \frac{101}{4} + \left(-\frac{1}{4} \times -1 \right) = \frac{101}{4} + \frac{1}{4} = \frac{102}{4} = 25.5 \][/tex]
4. Multiply the new remainder [tex]\(25.5\)[/tex] by [tex]\(-\frac{1}{4}\)[/tex] and add the next coefficient:
-quotient term: 25.5
[tex]\[ -\frac{125}{4} + \left(25.5 \times -\frac{1}{4} \right) = -\frac{125}{4} - 6.375 = -37.625 \][/tex]
5. Multiply the new remainder [tex]\(-37.625\)[/tex] by [tex]\(-\frac{1}{4}\)[/tex] and add the constant term:
-quotient term: -37.625
[tex]\[ 30 + \left(-37.625 \times -\frac{1}{4} \right) = 30 + 9.40625 = 39.40625 \][/tex]
So, the quotient of the division is [tex]\([-1, 25.5, -37.625]\)[/tex] and the remainder is [tex]\( 236.4375 \)[/tex].
Therefore, the final quotient and remainder are:
Quotient: [tex]\([-1, 25.5, -37.625]\)[/tex]
Remainder: [tex]\( 236.4375 \)[/tex]
### Step 1: Check if the divisor is in [tex]\( (x - r) \)[/tex] form
For a divisor to be in the form [tex]\( (x - r) \)[/tex], it should look something like [tex]\( x - r \)[/tex] where [tex]\( r \)[/tex] is a constant. In this case, the divisor is [tex]\( -\frac{1}{4}x + 6 \)[/tex].
Here the term with [tex]\( x \)[/tex] is [tex]\(-\frac{1}{4}x\)[/tex], not in the form [tex]\( x-r \)[/tex]. Thus, the divisor is not in the form [tex]\( (x - r) \)[/tex].
Answer: The divisor given is not in [tex]\( (x-r) \)[/tex] form. [tex]\( \square \)[/tex]
### Step 2: Identify the number of terms in the dividend
The dividend polynomial is [tex]\( -x^3 + \frac{101}{4}x^2 - \frac{125}{4}x + 30 \)[/tex].
Count the terms:
1. [tex]\( -x^3 \)[/tex]
2. [tex]\( \frac{101}{4}x^2 \)[/tex]
3. [tex]\( -\frac{125}{4}x \)[/tex]
4. [tex]\( 30 \)[/tex]
So, the dividend has 4 terms.
Answer: There are 4 terms in the dividend. [tex]\( \square \)[/tex]
### Step 3: Perform synthetic division
Given:
- Dividend: [tex]\( -x^3 + \frac{101}{4}x^2 - \frac{125}{4}x + 30 \)[/tex]
- Divisor: [tex]\( -\frac{1}{4}x + 6 \)[/tex]
Here are the coefficients of the dividend: [tex]\(-1, \frac{101}{4}, -\frac{125}{4}, 30\)[/tex].
And the constant term (leading coefficient of [tex]\( x \)[/tex] of the divisor) is [tex]\(-\frac{1}{4}\)[/tex].
#### Synthetic Division Process:
1. Write down the coefficients of the dividend: [tex]\([-1, \frac{101}{4}, -\frac{125}{4}, 30]\)[/tex].
2. The divisor coefficient is [tex]\(-\frac{1}{4}\)[/tex].
Now, start the synthetic division process:
- Initial remainder (first coefficient): -1
#### Process each coefficient step-by-step:
3. Multiply the initial remainder by divisor coefficient and add the next coefficient:
-quotient term: -1
[tex]\[ \frac{101}{4} + \left(-\frac{1}{4} \times -1 \right) = \frac{101}{4} + \frac{1}{4} = \frac{102}{4} = 25.5 \][/tex]
4. Multiply the new remainder [tex]\(25.5\)[/tex] by [tex]\(-\frac{1}{4}\)[/tex] and add the next coefficient:
-quotient term: 25.5
[tex]\[ -\frac{125}{4} + \left(25.5 \times -\frac{1}{4} \right) = -\frac{125}{4} - 6.375 = -37.625 \][/tex]
5. Multiply the new remainder [tex]\(-37.625\)[/tex] by [tex]\(-\frac{1}{4}\)[/tex] and add the constant term:
-quotient term: -37.625
[tex]\[ 30 + \left(-37.625 \times -\frac{1}{4} \right) = 30 + 9.40625 = 39.40625 \][/tex]
So, the quotient of the division is [tex]\([-1, 25.5, -37.625]\)[/tex] and the remainder is [tex]\( 236.4375 \)[/tex].
Therefore, the final quotient and remainder are:
Quotient: [tex]\([-1, 25.5, -37.625]\)[/tex]
Remainder: [tex]\( 236.4375 \)[/tex]
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