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To understand whether a number is a zero of a polynomial, we start by evaluating the polynomial at that number. Evaluating a polynomial means to substitute a given value for the variable and simplify it. For example, if we have a polynomial \( f(x) = 4x^5 - 2x^3 + 10 \) and want to evaluate it at \( x = 3 \), we substitute 3 for x:

• First, compute \( 4 \times 3^5 \). Since \( 3^5 = 243 \), we have \( 4 \times 243 = 972 \).
• Next, compute \( -2 \times 3^3 \). Since \( 3^3 = 27 \), we have \( -2 \times 27 = -54 \).
• Finally, add 10: \( 972 - 54 + 10 = 928 \).

Hence, \( f(3) = 928 \). Evaluating polynomials at various values lets us see where the polynomial equals zero, helping identify its zeros.

A zero of a polynomial is a value of x for which the polynomial equals zero. Mathematically, if \( f(c) = 0 \), then c is a zero of the polynomial \( f(x) \). These zeros are also known as roots of the polynomial. To identify a zero:

• Substitute the candidate value into the polynomial.
• Simplify the expression.
• Check if the result is zero.

For example, let’s consider the polynomial \( f(x) = 4x^5 - 2x^3 + 10 \) again and test whether \( 3 \) can be a zero. We have already calculated that \( f(3) = 928 \). Since 928 is not zero, 3 is not a zero of the polynomial. Repeat this method for any other candidates to determine if they are zeros of the polynomial.

The substitution method is a straightforward technique used to find the zeros of a polynomial. It involves substituting a candidate value into the polynomial and solving. If the result is zero, then the candidate is a zero of the polynomial. Here's how to use the substitution method step-by-step:

• Write down the polynomial.
• Select a candidate value to test.
• Substitute this value into the polynomial in place of the variable x.
• Simplify the expression completely.
• Check the result. If it equals zero, the candidate is a zero. If not, it isn't.

Let's apply this to the earlier problem with candidate \( n \) values 3, 5, \( \frac{5}{2} \), and \( \frac{3}{2} \). By substituting each of these into \( f(x) \), we found that none of them make the polynomial zero. Therefore, none of these values are zeros of the polynomial \( f(x) = 4x^5 - 2x^3 + 10 \). Using the substitution method, solving and understanding polynomials' zeros can be simplified significantly.