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Find the remainder when the polynomial [tex]$4x^4 - 6x^3 + 6x^2 - 1[tex]$[/tex] is divided by the polynomial [tex]$[/tex]2x^2 - 3$[/tex].

Sagot :

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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.
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