Answer :
To determine the greatest number of zeros that the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex] could have, we need to consider the degree of the polynomial.
1. Identify the Degree of the Polynomial:
- A polynomial's degree is the highest power of the variable [tex]\( x \)[/tex] that appears with a non-zero coefficient.
- In this polynomial, the highest power of [tex]\( x \)[/tex] is 6 (as in [tex]\( 7x^6 \)[/tex]), which makes 6 the degree of the polynomial.
2. Understanding Polynomial Degree and Zeros:
- According to the fundamental theorem of algebra, a polynomial of degree [tex]\( n \)[/tex] can have at most [tex]\( n \)[/tex] roots (or zeros). These roots could be real or complex.
- So, for a polynomial of degree 6, it can have up to 6 zeros.
3. Conclusion:
- Therefore, the greatest number of zeros that the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex] could have is 6.
So, the answer is 6.
1. Identify the Degree of the Polynomial:
- A polynomial's degree is the highest power of the variable [tex]\( x \)[/tex] that appears with a non-zero coefficient.
- In this polynomial, the highest power of [tex]\( x \)[/tex] is 6 (as in [tex]\( 7x^6 \)[/tex]), which makes 6 the degree of the polynomial.
2. Understanding Polynomial Degree and Zeros:
- According to the fundamental theorem of algebra, a polynomial of degree [tex]\( n \)[/tex] can have at most [tex]\( n \)[/tex] roots (or zeros). These roots could be real or complex.
- So, for a polynomial of degree 6, it can have up to 6 zeros.
3. Conclusion:
- Therefore, the greatest number of zeros that the polynomial [tex]\( f(x) = 7x^6 - 5x^5 + x \)[/tex] could have is 6.
So, the answer is 6.
