Answered

Get the most out of your questions with IDNLearn.com's extensive resources. Discover reliable and timely information on any topic from our network of experienced professionals.

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 :

GinnyAnsweridn
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.
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.