%C3%B7(x%2B2).jpeg "(2x4+5x3+8x+24)÷(x+2)")
Factoring and Dividing Polynomials: (2x⁴ + 5x³ + 8x + 24) ÷ (x + 2)
This article explores the process of dividing the polynomial (2x⁴ + 5x³ + 8x + 24) by the binomial (x + 2). We'll use polynomial long division to achieve this.
Understanding Polynomial Long Division
Polynomial long division is analogous to long division with numbers. It involves a systematic process of dividing a polynomial (the dividend) by another polynomial (the divisor) to obtain a quotient and a remainder.
Steps for Polynomial Long Division
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Set up the division:
_________ x + 2 | 2x⁴ + 5x³ + 0x² + 8x + 24 -
Divide the leading terms: Divide the leading term of the dividend (2x⁴) by the leading term of the divisor (x), which gives 2x³. Write this quotient above the dividend.
2x³ ______ x + 2 | 2x⁴ + 5x³ + 0x² + 8x + 24 -
Multiply the quotient by the divisor: Multiply the quotient (2x³) by the divisor (x + 2), which gives 2x⁴ + 4x³. Write this result below the dividend.
2x³ ______ x + 2 | 2x⁴ + 5x³ + 0x² + 8x + 24 2x⁴ + 4x³ -
Subtract: Subtract the result from the dividend.
2x³ ______ x + 2 | 2x⁴ + 5x³ + 0x² + 8x + 24 2x⁴ + 4x³ ------- x³ + 0x² -
Bring down the next term: Bring down the next term from the dividend (0x²).
2x³ ______ x + 2 | 2x⁴ + 5x³ + 0x² + 8x + 24 2x⁴ + 4x³ ------- x³ + 0x² + 8x -
Repeat steps 2-5: Repeat the process from step 2 using the new polynomial (x³ + 0x² + 8x).
- Divide the leading term (x³) by the leading term of the divisor (x), which gives x². Write this above the dividend.
- Multiply the quotient (x²) by the divisor (x + 2) which gives x³ + 2x².
- Subtract the result.
- Bring down the next term (24).
2x³ + x² _____ x + 2 | 2x⁴ + 5x³ + 0x² + 8x + 24 2x⁴ + 4x³ ------- x³ + 0x² + 8x x³ + 2x² ------- -2x² + 8x + 24 -
Continue the process: Continue repeating steps 2-5 until the degree of the remainder is less than the degree of the divisor.
- Divide the leading term (-2x²) by the leading term of the divisor (x), which gives -2x. Write this above the dividend.
- Multiply the quotient (-2x) by the divisor (x + 2) which gives -2x² - 4x.
- Subtract the result.
- Bring down the next term (24).
2x³ + x² - 2x ____ x + 2 | 2x⁴ + 5x³ + 0x² + 8x + 24 2x⁴ + 4x³ ------- x³ + 0x² + 8x x³ + 2x² ------- -2x² + 8x + 24 -2x² - 4x ------- 12x + 24 -
Final step:
- Divide the leading term (12x) by the leading term of the divisor (x), which gives 12. Write this above the dividend.
- Multiply the quotient (12) by the divisor (x + 2) which gives 12x + 24.
- Subtract the result.
2x³ + x² - 2x + 12 x + 2 | 2x⁴ + 5x³ + 0x² + 8x + 24 2x⁴ + 4x³ ------- x³ + 0x² + 8x x³ + 2x² ------- -2x² + 8x + 24 -2x² - 4x ------- 12x + 24 12x + 24 ------- 0
Result
Therefore, (2x⁴ + 5x³ + 8x + 24) ÷ (x + 2) = 2x³ + x² - 2x + 12 with a remainder of 0.
Conclusion
Polynomial long division is a powerful tool for dividing polynomials. It helps us understand the relationships between polynomials and factor them. The steps outlined above provide a systematic method for accurately dividing any polynomial by another.