Answer :
To determine the greatest number of zeros a polynomial could have, we look at the degree of the polynomial. The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] that appears in the polynomial.
For the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex], the highest power of [tex]\( x \)[/tex] is 6. This means the degree of the polynomial is 6.
The Fundamental Theorem of Algebra tells us that a polynomial of degree [tex]\( n \)[/tex] can have at most [tex]\( n \)[/tex] zeros. This includes real and complex zeros and counts multiplicity.
Therefore, since the degree of the polynomial [tex]\( f(x) \)[/tex] is 6, the greatest number of zeros it could have is 6.
For the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex], the highest power of [tex]\( x \)[/tex] is 6. This means the degree of the polynomial is 6.
The Fundamental Theorem of Algebra tells us that a polynomial of degree [tex]\( n \)[/tex] can have at most [tex]\( n \)[/tex] zeros. This includes real and complex zeros and counts multiplicity.
Therefore, since the degree of the polynomial [tex]\( f(x) \)[/tex] is 6, the greatest number of zeros it could have is 6.
