graphing polynomial functions\nat which root does the graph of $f(x) = (x + 4)^6(x + 7)^5$ cross the…
graphing polynomial functions\nat which root does the graph of $f(x) = (x + 4)^6(x + 7)^5$ cross the $x$-axis?\n7 -7\n-4 4
Answer
Explanation:
Step1: Recall root and multiplicity rule
For a polynomial ( f(x) = (x - a)^n ), the root is ( x = a ), and if ( n ) is odd, the graph crosses the ( x )-axis at ( x = a ); if ( n ) is even, the graph touches the ( x )-axis (bounces) at ( x = a ).
Step2: Find roots of ( f(x) = (x + 4)^6(x + 7)^5 )
Rewrite the factors: ( (x + 4)^6=(x - (-4))^6 ) and ( (x + 7)^5=(x - (-7))^5 ). So the roots are ( x=-4 ) (multiplicity 6, even) and ( x = -7 ) (multiplicity 5, odd).
Step3: Determine crossing root
Since the multiplicity of ( x=-7 ) is odd (5), the graph crosses the ( x )-axis at ( x=-7 ). The multiplicity of ( x=-4 ) is even (6), so the graph touches (bounces) at ( x=-4 ). The other options (7 and 4) are not roots (since ( (x - 7) ) or ( (x - 4) ) are not factors).