decompose the polynomial x^3 -13x^2 +43x-25 into two factors, given that x-5 is a factor
To find the other factor, we can perform long division.
x^2 - 8x + 5
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x - 5 | x^3 - 13x^2 + 43x - 25
- (x^3 - 5x^2)
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- 8x^2 + 43x
- (-8x^2 + 40x)
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3x - 25
- (3x - 15)
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- 10
Therefore, we have (x^3 - 13x^2 + 43x - 25) = (x - 5)(x^2 - 8x + 5).
Thus, the polynomial can be decomposed as: x^3 - 13x^2 + 43x - 25 = (x - 5)(x^2 - 8x + 5).
To decompose the polynomial x^3 - 13x^2 + 43x - 25 into two factors, we need to use the factor theorem.
Given that x - 5 is a factor, we can use polynomial long division or synthetic division to divide the given polynomial by x - 5 to find the other factor.
Let's perform polynomial long division to find the other factor:
x^2 - 8x + 5
---------------------
x - 5 │ x^3 - 13x^2 + 43x - 25
- (x^3 - 5x^2)
----------------
- 8x^2 + 43x
- (-8x^2 + 40x)
----------------
3x - 25
- (3x - 15)
--------------
-10
The quotient is x^2 - 8x + 5, and the remainder is -10. Therefore, we can rewrite the polynomial as:
(x - 5)(x^2 - 8x + 5) - 10
So, the decomposed form of the polynomial x^3 - 13x^2 + 43x - 25 is (x - 5)(x^2 - 8x + 5) - 10.
To decompose the polynomial x^3 -13x^2 +43x-25 into two factors, we will use synthetic division to divide the polynomial by the given factor, x-5.
Step 1: Set up synthetic division
5 | 1 -13 43 -25
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Step 2: Bring down the first coefficient (1) to the bottom row.
5 | 1 -13 43 -25
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1
Step 3: Multiply the divisor (5) by the number at the bottom of the column (1), and write the result under the next coefficient.
5 | 1 -13 43 -25
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1
-5
Step 4: Add the numbers in the right column.
5 | 1 -13 43 -25
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1
-5
------
-4
Step 5: Repeat steps 2-4 until all the coefficients have been brought down.
5 | 1 -13 43 -25
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1 0 43
-5
------
-4 -13 30
20
------
16 7 5
Step 6: Write the final result and the remainder.
The final result of the synthetic division is: 1x^2 + 0x + 43 and the remainder is 16.
Therefore, the polynomial x^3 -13x^2 +43x-25 can be factored as (x - 5)(x^2 + 0x + 43) + 16.