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Given a polynomial and one of its factors, drag the remaining factors of the polynomial f(x)=x^3+3x^2-4;x-1

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Given a polynomial and one of its factors, drag the remaining factors of the polynomial f(x)=x^3+3x^2-4;x-1

Given a polynomial and one of its factors, drag the remaining factors of the polynomial f(x)=x^3+3x^2-4;x-1

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KieranProfessional · Tutor for 6 years

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<p> <br /><br />Solution:<br />1. Substitute x=1 into f(x) = x³ + 3x² - 4:<br /><br />Replace x = 1 into the given function:<br /> f(1) = (1**3) + 3*(1**2) - 4 = 1 + 3 - 4 = 0<br />Thus f(1) = 0 verifying that x - 1 is indeed a factor of the given polynomial.<br /><br />2. Factor the given polynomial:<br /><br />We divide the original polynomial by the given factor(x-1) using synthetic division:<br />```<br /> 1 | 1 3 0 -4<br /> | 1 4 4<br /> |_________<br /> 1 4 4 0<br />```<br />=> The quadratic polynomial f(x)/ (x - 1) = x² + 4x + 4<br /><br />3. Find roots of the quadratic and thereby identifying remaining factors:<br /><br />We'll factorize f(x)/ (x - 1) . This can be written as: <br />(x² + 4x + 4) = (x + 2)(x + 2)<br />Every root of this quadratic represents a factor x - root :<br />Hence, the remaining factors are x + 2, x + 2.<br /><br />In conclusion: `f(x)=x³+3x²-4` fully factors as `(x-1) * (x+2) * (x+2)`, illustratively broken down as: [x - 1], [x + 2], [x + 2].

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<p> This math problem is in the domain of polynomial factoring.<br />Given that `x-1` is a factor of `f(x)=x³+3x²-4`, and by the Factor Theorem if `x=a` is a root of a polynomial, then `x-a` is a factor of that polynomial.<br /><br />Consequently, if we substitute x=1 in the given expression `f(x) = x³ + 3x² - 4`, if `f(1)` equals 0, THEN ONLY `x - 1` is a factor of the given expression.<br /><br />So, first, we need to substitute x=1 into `f(x)` to confirm whether `x - 1` is truly a factor.<br /><br />If `x - 1` is indeed a factor, we have found one of the roots of the given polynomial equation, and divide the polynomial by this confirmed factor will give us the quadratic expression. <br /><br />Then we can find the final factors by solving the quadratic expression using the quadratic formula. Any roots from the quadratic can be made into factors by `x - rootVal`.
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