Math Problem Statement

(4x^5-6x^4+5x^2-x-10)÷(2x^2+3)

Solution

To divide the polynomial (4x56x4+5x2x10)(4x^5 - 6x^4 + 5x^2 - x - 10) by (2x2+3)(2x^2 + 3), we'll use polynomial long division. Here are the steps:

Step 1: Set up the division

We write the division as:

4x56x4+5x2x102x2+3\frac{4x^5 - 6x^4 + 5x^2 - x - 10}{2x^2 + 3}

Step 2: Divide the first term of the dividend by the first term of the divisor

Divide 4x54x^5 by 2x22x^2:

4x52x2=2x3\frac{4x^5}{2x^2} = 2x^3

This will be the first term of the quotient.

Step 3: Multiply the entire divisor by this term and subtract

Multiply 2x32x^3 by the divisor 2x2+32x^2 + 3:

(2x3)(2x2+3)=4x5+6x3(2x^3)(2x^2 + 3) = 4x^5 + 6x^3

Now subtract this from the original polynomial:

(4x56x4+5x2x10)(4x5+6x3)=6x46x3+5x2x10(4x^5 - 6x^4 + 5x^2 - x - 10) - (4x^5 + 6x^3) = -6x^4 - 6x^3 + 5x^2 - x - 10

Step 4: Repeat the process

Now, divide the new leading term 6x4-6x^4 by 2x22x^2:

6x42x2=3x2\frac{-6x^4}{2x^2} = -3x^2

Multiply 3x2-3x^2 by the divisor 2x2+32x^2 + 3:

(3x2)(2x2+3)=6x49x2(-3x^2)(2x^2 + 3) = -6x^4 - 9x^2

Subtract this from the current polynomial:

(6x46x3+5x2x10)(6x49x2)=6x3+14x2x10(-6x^4 - 6x^3 + 5x^2 - x - 10) - (-6x^4 - 9x^2) = -6x^3 + 14x^2 - x - 10

Step 5: Continue dividing

Next, divide 6x3-6x^3 by 2x22x^2:

6x32x2=3x\frac{-6x^3}{2x^2} = -3x

Multiply 3x-3x by the divisor 2x2+32x^2 + 3:

(3x)(2x2+3)=6x39x(-3x)(2x^2 + 3) = -6x^3 - 9x

Subtract this from the polynomial:

(6x3+14x2x10)(6x39x)=14x2+8x10(-6x^3 + 14x^2 - x - 10) - (-6x^3 - 9x) = 14x^2 + 8x - 10

Step 6: Divide again

Divide 14x214x^2 by 2x22x^2:

14x22x2=7\frac{14x^2}{2x^2} = 7

Multiply 77 by the divisor 2x2+32x^2 + 3:

7(2x2+3)=14x2+217(2x^2 + 3) = 14x^2 + 21

Subtract this from the polynomial:

(14x2+8x10)(14x2+21)=8x31(14x^2 + 8x - 10) - (14x^2 + 21) = 8x - 31

Final Answer:

The quotient is:

2x33x23x+72x^3 - 3x^2 - 3x + 7

And the remainder is:

8x312x2+3\frac{8x - 31}{2x^2 + 3}

So, the complete expression is:

4x56x4+5x2x102x2+3=2x33x23x+7+8x312x2+3\frac{4x^5 - 6x^4 + 5x^2 - x - 10}{2x^2 + 3} = 2x^3 - 3x^2 - 3x + 7 + \frac{8x - 31}{2x^2 + 3}

Would you like any additional details or explanations?

Related Questions:

  1. How do you perform synthetic division on polynomials?
  2. What are the applications of polynomial division in calculus?
  3. How can you determine if a polynomial is divisible by another polynomial without a remainder?
  4. What is the significance of the remainder in polynomial division?
  5. Can polynomial division be used to find roots of polynomials?

Tip:

Always double-check each step in polynomial division to avoid mistakes, especially when subtracting polynomials during the process.

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Math Problem Analysis

Mathematical Concepts

Polynomial Division
Long Division of Polynomials

Formulas

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Theorems

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Suitable Grade Level

High School