Question Factor the following polynomial completely, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = x^4 + x^3 + 2x-4 }
| Foundations
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| 1) What does the Rational Zeros Theorem say about the possible zeros?
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| 2) How do you check if a possible zero is actually a zero?
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| 3) How do you find the rest of the zeros?
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| Answer:
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| 1) The possible divisors can be found by finding the factors of -10, in a list, and the factors of 1, in a second list. Then write down all the fractions with numerators from the first list and denominators from the second list.
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| 2) Use synthetic division, or plug a possible zero into the function. If you get 0, you have found a zero.
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| 3) After your reduce the polynomial with synthetic division, try and find another zero from the list you made in part a). Once you reach a degree 2 polynomial you can finish the problem with the quadratic formula.
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| Step 1:
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| First, we use the Rational Zeros Theorem to note that the possible zeros are: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\pm 1, \pm 2, \pm 4 \}}
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| Step 2:
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| Now we start checking which of the possible roots are actually roots. We find that 1 is a zero, and apply either synthetic division or long division to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^4 + x^3 +2x - 4 = (x - 1)(x^3 +2x^2 + 2x +4)}
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| Step 3:
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| We continue checking zeros and find that -2 is a zero. Applying synthetic division or long division we can simplify the polynomial down to:
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| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^4 +x^3+ 2x -4 = (x - 1)(x + 2)(x^2 + 2)}
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| Step 4:
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| Now we can finish the problem by applying the quadratic formula or just finding the roots of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{2}+2}
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| Final Answer:
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{4}+x^{3}+2x-4=(x-1)(x+2)(x-{\sqrt {2}}i)(x+{\sqrt {2}}i)}
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