Question

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Use the synthetic division to show that c is a zero of f(x).
f(x) = 4x^3 - 6x^2 + 8x - 3; c = 1/2

f(1/2) = ________________

Answer 1

`f\left((1)/(2)\right)=0`

Explanation:

Given polynomial:

`f(x) = 4x^3 - 6x^2 + 8x - 3`

The zeros of a function are the values of the variable that make the function zero. Therefore, if c = 1/2 is a zero of f(x), then f(1/2) = 0.

According to the factor theorem, if f(x) is a polynomial, and f(c) = 0, then (x - c) is a factor if f(x). Given c = 1/2, then (x - 1/2) is a factor of f(x).

Therefore, to show that c is a zero of f(x), divide f(x) by (x - 1/2) using synthetic division.

`\textsf{Dividend:}\quad 4x^3 - 6x^2 + 8x - 3`

`\textsf{Divisor:} \quad \left(x-(1)/(2)\right)`

Perform Synthetic Division

Place the zero "c" in the division box.

Write the coefficients of the dividend in descending order.

(Note: As no terms are missing, we do not need to use any zeros to fill in missing terms).

`\begin{array}c\vphantom{\frac12}(1)/(2) & 4 & -6 & 8 & -3\\\cline{1-1}\end{array}`

Bring the leading coefficient straight down:

`\begin{array}rrrr\vphantom{\frac12}(1)/(2)&4&-6&8&-3\\\cline{1-1}&\downarrow &&&\\\cline{2-5}&4&&&\end{array}`

Multiply the number you brought down with the number in the division box and put the result in the next column (under the -6):

`\begin{array}c\vphantom{\frac12}(1)/(2)&4&-6&8&-3\\\cline{1-1}&\downarrow &2&&\\\cline{2-5}&4&&&\end{array}`

Add the two numbers together and put the result in the bottom row:

`\begin{array}rrrr\vphantom{\frac12}(1)/(2)&4&-6&8&-3\\\cline{1-1}&\downarrow &2&&\\\cline{2-5}&4&-4&&\end{array}`

Repeat:

`\begin{array}rrrr\vphantom{\frac12}(1)/(2)&4&-6&8&-3\\\cline{1-1}&\downarrow &2&-2&\\\cline{2-5}&4&-4&6&\end{array}`

`\begin{array}c\vphantom{\frac12}(1)/(2)&4&-6&8&-3\\\cline{1-1}&\downarrow &2&-2&3\\\cline{2-5}&4&-4&6&0\end{array}`

The bottom row (except the last number) gives the coefficients of the quotient. The degree of the quotient is one less than that of the dividend.

The last number in the bottom row is the remainder.

Therefore:

• Quotient: 4x² - 4x + 6
• Remainder: 0

As the remainder is f(c) then:

`f\left((1)/(2)\right)=0`