What is the result when 2x^3 -9x^2 +11x-6 is divided by x-3?
Could I have some help solving this? I would like to know the steps I should take to get the correct answer.
Thank you!
To divide the polynomial 2x^3 - 9x^2 + 11x - 6 by binomial x - 3, you can use long division method. Here are the step-by-step instructions:
Step 1: Write the polynomial in descending order.
2x^3 - 9x^2 + 11x - 6
Step 2: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x).
(2x^3 / x) = 2x^2
Step 3: Multiply the entire divisor (x - 3) by the quotient obtained in step 2 (2x^2).
(2x^2 * (x - 3)) = 2x^3 - 6x^2
Step 4: Subtract the product obtained in step 3 from the dividend.
(2x^3 - 9x^2 + 11x - 6) - (2x^3 - 6x^2) = -3x^2 + 11x - 6
Step 5: Bring down the next term from the original dividend (-3x^2).
-3x^2
Step 6: Divide the new leading term of the resulting polynomial (-3x^2) by the leading term of the divisor (x).
(-3x^2 / x) = -3x
Step 7: Multiply the entire divisor (x - 3) by the quotient obtained in step 6 (-3x).
(-3x * (x - 3)) = -3x^2 + 9x
Step 8: Subtract the product obtained in step 7 from the current resulting polynomial.
(-3x^2 + 11x - 6) - (-3x^2 + 9x) = 2x - 6
Step 9: Bring down the next term from the original dividend (2x).
2x
Step 10: Divide the new leading term of the resulting polynomial (2x) by the leading term of the divisor (x).
(2x / x) = 2
Step 11: Multiply the entire divisor (x - 3) by the quotient obtained in step 10 (2).
(2 * (x - 3)) = 2x - 6
Step 12: Subtract the product obtained in step 11 from the current resulting polynomial.
(2x - 6) - (2x - 6) = 0
Since the resulting polynomial is 0, we have no remainder.
Therefore, the result of dividing 2x^3 - 9x^2 + 11x - 6 by x - 3 is 2x^2 - 3x + 2.
To divide the polynomial 2x^3 - 9x^2 + 11x - 6 by x - 3, you can use polynomial long division. Here are the steps to follow:
1. Arrange the polynomial in descending order of exponents:
2x^3 - 9x^2 + 11x - 6
2. Divide the first term of the polynomial (2x^3) by the first term of the divisor (x), which gives you 2x^2 as the quotient. Write this above the division symbol.
2x^2
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x - 3 | 2x^3 - 9x^2 + 11x - 6
3. Multiply the divisor (x - 3) by the quotient (2x^2) and write the result under the first three terms of the dividend (2x^3 - 9x^2).
2x^2 * (x - 3) = 2x^3 - 6x^2
4. Subtract the result from step 3 from the dividend.
(2x^3 - 9x^2 + 11x - 6) - (2x^3 - 6x^2) = (-9x^2 + 11x - 6 + 6x^2)
5. Bring down the next term from the dividend, which is 11x, and add it to the result of the subtraction.
(-9x^2 + 11x - 6 + 6x^2) + 11x = (-9x^2 + 6x^2 + 11x + 11x - 6)
6. Repeat steps 2-5 until all terms in the dividend have been divided.
The next step would be to multiply the divisor (x - 3) by the new quotient obtained in step 5 and perform the subtraction again. Continue this process until you have divided all terms in the dividend.
7. The final result will be the quotient obtained after dividing all terms completely. The remainder of the division would be zero if the division is exact.
Please note that I have performed the division for you and the result is:
Quotient: 2x^2 + 5x + 17
Remainder: 39
So, the result of dividing 2x^3 - 9x^2 + 11x - 6 by x - 3 is 2x^2 + 5x + 17 with a remainder of 39.
a little synthetic division shows that
2x^3 -9x^2 +11x-6 = (x-3)(2x^2-3x+2)
If you have not studied synthetic division yet, google polynomial long division for examples. Of course, you can also google synthetic division...