Problem

Factor \(x^{2}-8x+16=0\) as a perfect-square trinomial. Verify the middle term from the two square bases, then state the repeated zero and its multiplicity.

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A perfect-square trinomial represents two identical linear factors, not two distinct zeros. We’ll verify the signed middle term from the endpoint square roots, rewrite the trinomial as one squared binomial, solve that repeated factor, and connect its multiplicity to a tangent x-intercept.

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end roots=a=x,b=4; middle check=−2ab=−8x; squared factor=(x−4)²=0; zeros=4 and 4; multiplicity=1 each; graph meaning=two distinct x-intercepts

end roots=a=x,b=4; middle check=−2ab=−8x; squared factor=(x−4)²=0; repeated zero=-4; multiplicity2; graph meaning=tangent at the copied value

Roots x and 4; middle check +2(x)(4) = +8x; squared factor (x + 4)² = 0; repeated zero x = −4; multiplicity 2; expansion x² + 8x + 16.

end roots=a=x,b=4; middle check=-2ab=-8x; factored=(x−4)²=0; repeated zero=x=4; multiplicity=2; graph meaning=tangent x-intercept(4,0)

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Factor \(x^{2}-8x+16=0\) as a perfect-square trinomial. Verify the middle term from the two square bases, then state the repeated zero and its multiplicity.

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