Factor the polynomial P(x)=x4−2x3+8x−16 and use the factored form to find the zeros. Then sketch the graph.
Since the function has an even degree of 4 and a positive leading coefficient, its end behaviour is y→∞ as x→−∞ and y→∞ as x→∞. To find the x intercepts (or zeros), we set y=0.
0=x4−2x3+8x−160=x4−2x3+(8x−16)Group terms0=x3(x−2)+8(x−2)Factor out x−20=(x3+8)(x−2)Factor out x3+8
By zero product property, we have
x3+8=0 and x−2=0
Thus, the x-intercept are x=−2 and x=2
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