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Asked Dec 2, 2019
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4. Use the Rational Zeros Theorem to find all the zeros of the function and use the zeros to factor the function
f(x) 2x4-x3 - 5x2 2x+ 2
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4. Use the Rational Zeros Theorem to find all the zeros of the function and use the zeros to factor the function f(x) 2x4-x3 - 5x2 2x+ 2

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Expert Answer

Step 1

According to the rational roots theorem, if  f(x) is a polynomial with integer coefficients, then every zero of the polynomial has the form p/q where p is a factor of the constant term of  f(x) and q is a factor of the leading coefficient.

Obtain all the possible rational zeroes as follows.

f(x) 2x-x-5r2 +2x+2
The constant tem is 2 and the factors of 2 are p 1,+2
The leading coefficient is 2 and the factors of 2 are q =12
р
The list of all possible rational zeroes are
+1,t,+2
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f(x) 2x-x-5r2 +2x+2 The constant tem is 2 and the factors of 2 are p 1,+2 The leading coefficient is 2 and the factors of 2 are q =12 р The list of all possible rational zeroes are +1,t,+2

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Step 2

Determine which of the possible zeros are actual zeros by substituting these va...

f(-1)= 2(-)()-5(-1+2(-1)+2=-2
f()2(1)(-5(1+2(1)+2-0
x1 is a zero.
(x-1) is a factor of f(x)
= (x-1)r-r -5x 2x +2
f(x)
x-1
(x-1)(2x+x-4x-2)
2x3x4x-2
x-12r-x-5x+2r+2
2r-213
3-5x
-4x2+2x
4x24x
-2r +2
-2r+2
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f(-1)= 2(-)()-5(-1+2(-1)+2=-2 f()2(1)(-5(1+2(1)+2-0 x1 is a zero. (x-1) is a factor of f(x) = (x-1)r-r -5x 2x +2 f(x) x-1 (x-1)(2x+x-4x-2) 2x3x4x-2 x-12r-x-5x+2r+2 2r-213 3-5x -4x2+2x 4x24x -2r +2 -2r+2

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Algebra