Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = 5x⁴ + 2x³ -x+3; k=2/5

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = 5x^4 + 2x^3 + 0x^2 - 1x + 3. The coefficients are 5, 2, 0, -1, and 3.
2. Set up synthetic division using k = \frac{2}{5}. Write the coefficients in a row and place k to the left.
3. Bring down the first coefficient (5) as it is. Multiply this number by k (\frac{2}{5}) and write the result under the next coefficient.
4. Add the second coefficient (2) and the number just written. Write the sum below the line. Repeat the multiply and add process for all coefficients.
5. The last number you get after completing synthetic division is the remainder, which equals ƒ(k). If this remainder is 0, then k is a zero of the polynomial; otherwise, it is not.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - k). It simplifies the long division process by using only the coefficients of the polynomial, making it faster and less error-prone. This method helps determine if k is a root by checking if the remainder is zero.

Polynomial Functions and Zeros

A zero of a polynomial function is a value of x that makes the function equal to zero. If substituting k into the polynomial yields zero, then k is a root or zero of the polynomial. Identifying zeros is essential for factoring and graphing polynomial functions.

Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value k means substituting k into the polynomial and calculating the result. If the result is not zero, it gives the value of ƒ(k), which indicates that k is not a zero of the polynomial. Synthetic division can also provide this value as the remainder.