3. Factor x^4-3x^3-11x^2+3x+10 completely using the factor theorem. (5 marks)
SHOW ME ALL THE STEPS PLEASE.
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To factor the polynomial x^4 - 3x^3 - 11x^2 + 3x + 10 completely using the factor theorem, we need to follow these steps:
Step 1: Verify possible rational roots
First, we need to find the possible rational roots of the polynomial. According to the rational root theorem, the possible rational roots are of the form p/q, where p is a factor of the constant term (10) and q is a factor of the leading coefficient (1). The factors of 10 are ±1, ±2, ±5, ±10, and the factors of 1 are just ±1. So, the possible rational roots are ±1, ±2, ±5, ±10.
Step 2: Test the possible rational roots
Next, we will test each of the possible rational roots by substituting them into the polynomial and checking if it equals to zero.
Let's start by testing the roots: 1, -1, 2, -2, 5, -5, 10, -10
For x = 1:
(1)^4 - 3(1)^3 - 11(1)^2 + 3(1) + 10 = 1 - 3 - 11 + 3 + 10 = 0
Therefore, x - 1 is a factor of x^4 - 3x^3 - 11x^2 + 3x + 10.
Step 3: Perform long division
Now we divide the polynomial by x - 1 to obtain the remaining factor.
x^3 - 2x^2 - 9x - 10
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x - 1 | x^4 - 3x^3 - 11x^2 + 3x + 10
- x^4 + x^3
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- 4x^3 - 11x^2 + 3x
+ 4x^3 - 4x^2
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- 7x^2 + 3x + 10
+ 7x^2 - 7x
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- 4x + 10
+ 4x - 4
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6
The quotient is x^3 - 2x^2 - 9x - 10, and the remainder is 6.
Step 4: Continue factoring
Since there is a remainder of 6, the polynomial x^3 - 2x^2 - 9x - 10 cannot be factored further using rational roots.
So the complete factorization of x^4 - 3x^3 - 11x^2 + 3x + 10 using the factor theorem is:
(x - 1)(x^3 - 2x^2 - 9x - 10)