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Let's solve the problem of dividing the polynomial [tex]\( P(x) = 20x^4 + 16x^3 + 9x^2 \)[/tex] by the polynomial [tex]\( D(x) = 4x^2 + 1 \)[/tex] using polynomial division.
### 1. Write Down Polynomials
First, let's rewrite the polynomials [tex]\( P(x) \)[/tex] and [tex]\( D(x) \)[/tex] clearly:
[tex]\[ P(x) = 20x^4 + 16x^3 + 9x^2 + 0x + 0 \][/tex]
[tex]\[ D(x) = 4x^2 + 0x + 1 \][/tex]
### 2. Setup the Division
We need to find the quotient [tex]\( Q(x) \)[/tex] and the remainder [tex]\( R(x) \)[/tex] such that:
[tex]\[ P(x) = D(x) \cdot Q(x) + R(x) \][/tex]
### 3. Perform the Division (Long Division Method)
1. First Term Calculation:
- Divide the leading term of [tex]\( P(x) \)[/tex] by the leading term of [tex]\( D(x) \)[/tex]:
[tex]\[ \frac{20x^4}{4x^2} = 5x^2 \][/tex]
- The first term of [tex]\( Q(x) \)[/tex] is [tex]\( 5x^2 \)[/tex].
- Multiply [tex]\( D(x) \)[/tex] by [tex]\( 5x^2 \)[/tex] and subtract from [tex]\( P(x) \)[/tex]:
[tex]\[ (20x^4 + 16x^3 + 9x^2) - (20x^4 + 0x^3 + 5x^2) = 16x^3 + 4x^2 \][/tex]
2. Second Term Calculation:
- Divide the new leading term by the leading term of [tex]\( D(x) \)[/tex]:
[tex]\[ \frac{16x^3}{4x^2} = 4x \][/tex]
- The second term of [tex]\( Q(x) \)[/tex] is [tex]\( 4x \)[/tex].
- Multiply [tex]\( D(x) \)[/tex] by [tex]\( 4x \)[/tex] and subtract from the result obtained:
[tex]\[ (16x^3 + 4x^2) - (16x^3 + 0x^2 + 4x) = 4x^2 - 4x \][/tex]
3. Third Term Calculation:
- Divide the new leading term by the leading term of [tex]\( D(x) \)[/tex]:
[tex]\[ \frac{4x^2}{4x^2} = 1 \][/tex]
- The third term of [tex]\( Q(x) \)[/tex] is [tex]\( 1 \)[/tex].
- Multiply [tex]\( D(x) \)[/tex] by [tex]\( 1 \)[/tex] and subtract from the result obtained:
[tex]\[ (4x^2 - 4x) - (4x^2 + 0x + 1) = -4x - 1 \][/tex]
Now we've completed dividing [tex]\( P(x) \)[/tex] by [tex]\( D(x) \)[/tex].
### 4. Result
- The quotient [tex]\( Q(x) \)[/tex] is the combination of terms we calculated: [tex]\( 5x^2 + 4x + 1 \)[/tex].
- The remainder [tex]\( R(x) \)[/tex] is [tex]\( -4x - 1 \)[/tex].
### 5. Write the Final Answer
The quotient and remainder of dividing [tex]\( P(x) = 20x^4 + 16x^3 + 9x^2 \)[/tex] by [tex]\( D(x) = 4x^2 + 1 \)[/tex] are:
[tex]\[ Q(x) = 5x^2 + 4x + 1 \][/tex]
[tex]\[ R(x) = -4x - 1 \][/tex]
So, the final answer is:
[tex]\[ \boxed{\left( Q(x) = 5x^2 + 4x + 1, \; R(x) = -4x - 1 \right)} \][/tex]
### 1. Write Down Polynomials
First, let's rewrite the polynomials [tex]\( P(x) \)[/tex] and [tex]\( D(x) \)[/tex] clearly:
[tex]\[ P(x) = 20x^4 + 16x^3 + 9x^2 + 0x + 0 \][/tex]
[tex]\[ D(x) = 4x^2 + 0x + 1 \][/tex]
### 2. Setup the Division
We need to find the quotient [tex]\( Q(x) \)[/tex] and the remainder [tex]\( R(x) \)[/tex] such that:
[tex]\[ P(x) = D(x) \cdot Q(x) + R(x) \][/tex]
### 3. Perform the Division (Long Division Method)
1. First Term Calculation:
- Divide the leading term of [tex]\( P(x) \)[/tex] by the leading term of [tex]\( D(x) \)[/tex]:
[tex]\[ \frac{20x^4}{4x^2} = 5x^2 \][/tex]
- The first term of [tex]\( Q(x) \)[/tex] is [tex]\( 5x^2 \)[/tex].
- Multiply [tex]\( D(x) \)[/tex] by [tex]\( 5x^2 \)[/tex] and subtract from [tex]\( P(x) \)[/tex]:
[tex]\[ (20x^4 + 16x^3 + 9x^2) - (20x^4 + 0x^3 + 5x^2) = 16x^3 + 4x^2 \][/tex]
2. Second Term Calculation:
- Divide the new leading term by the leading term of [tex]\( D(x) \)[/tex]:
[tex]\[ \frac{16x^3}{4x^2} = 4x \][/tex]
- The second term of [tex]\( Q(x) \)[/tex] is [tex]\( 4x \)[/tex].
- Multiply [tex]\( D(x) \)[/tex] by [tex]\( 4x \)[/tex] and subtract from the result obtained:
[tex]\[ (16x^3 + 4x^2) - (16x^3 + 0x^2 + 4x) = 4x^2 - 4x \][/tex]
3. Third Term Calculation:
- Divide the new leading term by the leading term of [tex]\( D(x) \)[/tex]:
[tex]\[ \frac{4x^2}{4x^2} = 1 \][/tex]
- The third term of [tex]\( Q(x) \)[/tex] is [tex]\( 1 \)[/tex].
- Multiply [tex]\( D(x) \)[/tex] by [tex]\( 1 \)[/tex] and subtract from the result obtained:
[tex]\[ (4x^2 - 4x) - (4x^2 + 0x + 1) = -4x - 1 \][/tex]
Now we've completed dividing [tex]\( P(x) \)[/tex] by [tex]\( D(x) \)[/tex].
### 4. Result
- The quotient [tex]\( Q(x) \)[/tex] is the combination of terms we calculated: [tex]\( 5x^2 + 4x + 1 \)[/tex].
- The remainder [tex]\( R(x) \)[/tex] is [tex]\( -4x - 1 \)[/tex].
### 5. Write the Final Answer
The quotient and remainder of dividing [tex]\( P(x) = 20x^4 + 16x^3 + 9x^2 \)[/tex] by [tex]\( D(x) = 4x^2 + 1 \)[/tex] are:
[tex]\[ Q(x) = 5x^2 + 4x + 1 \][/tex]
[tex]\[ R(x) = -4x - 1 \][/tex]
So, the final answer is:
[tex]\[ \boxed{\left( Q(x) = 5x^2 + 4x + 1, \; R(x) = -4x - 1 \right)} \][/tex]
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