Use polynomial long division to perform the indi-cated division. Write the polynomial in the form p(x) = d(x)q(x) + r(x).(a) (5x4 − 3x3 + 2x2 − 1) ÷ (x2 + 4)
Question
Use polynomial long division to perform the indi-cated division. Write the polynomial in the form p(x) = d(x)q(x) + r(x).(a) (5x4 − 3x3 + 2x2 − 1) ÷ (x2 + 4)
Solution
Sure, let's perform the polynomial long division step by step.
Step 1: Write the division in long division format.
_______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1
Step 2: Divide the first term of the dividend (5x⁴) by the first term of the divisor (x²) to get the first term of the quotient (5x²). Write this above the line.
5x²
_______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1
Step 3: Multiply the divisor (x² + 4) by the first term of the quotient (5x²) and subtract the result from the dividend.
5x²
_______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1
- (5x⁴ + 20x²)
-----------------
-3x³ - 18x² - 1
Step 4: Repeat the process. Divide the first term of the new dividend (-3x³) by the first term of the divisor (x²) to get the next term of the quotient (-3x). Write this above the line.
5x² - 3x
_______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1
- (5x⁴ + 20x²)
-----------------
-3x³ - 18x² - 1
- (-3x³ - 12x)
-----------------
-6x² + 12x - 1
Step 5: Repeat the process again. Divide the first term of the new dividend (-6x²) by the first term of the divisor (x²) to get the next term of the quotient (-6). Write this above the line.
5x² - 3x - 6
_______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1
- (5x⁴ + 20x²)
-----------------
-3x³ - 18x² - 1
- (-3x³ - 12x)
-----------------
-6x² + 12x - 1
- (-6x² - 24)
-----------------
12x + 23
Step 6: Since the degree of the new dividend (12x + 23) is less than the degree of the divisor (x² + 4), we stop here. The new dividend is the remainder.
So, the polynomial in the form p(x) = d(x)q(x) + r(x) is:
5x⁴ - 3x³ + 2x² - 1 = (x² + 4)(5x² - 3x - 6) + (12x + 23)
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