Answer :
To find the greatest number of zeros a polynomial can have, you need to look at its degree. The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] in its expression.
Let's analyze the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex]:
1. Identify the terms: The terms in this polynomial are [tex]\( 7x^6 \)[/tex], [tex]\( -5x^5 \)[/tex], and [tex]\( x \)[/tex].
2. Determine the degree: The degree of a polynomial is determined by the highest power of [tex]\( x \)[/tex]. In this polynomial, the term with the highest power of [tex]\( x \)[/tex] is [tex]\( 7x^6 \)[/tex]. Therefore, the degree of the polynomial is 6.
3. Number of zeros: A fundamental theorem of algebra states that a polynomial of degree [tex]\( n \)[/tex] can have at most [tex]\( n \)[/tex] zeros. These zeros can be real or complex numbers.
So, for the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex], which is of degree 6, the greatest number of zeros it can have is 6.
Let's analyze the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex]:
1. Identify the terms: The terms in this polynomial are [tex]\( 7x^6 \)[/tex], [tex]\( -5x^5 \)[/tex], and [tex]\( x \)[/tex].
2. Determine the degree: The degree of a polynomial is determined by the highest power of [tex]\( x \)[/tex]. In this polynomial, the term with the highest power of [tex]\( x \)[/tex] is [tex]\( 7x^6 \)[/tex]. Therefore, the degree of the polynomial is 6.
3. Number of zeros: A fundamental theorem of algebra states that a polynomial of degree [tex]\( n \)[/tex] can have at most [tex]\( n \)[/tex] zeros. These zeros can be real or complex numbers.
So, for the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex], which is of degree 6, the greatest number of zeros it can have is 6.
