Question
When you use synthetic division to divide x^4+x^3−2x^2+2x−5 by 2x+1, what is the value of the divisor?
Answers
GPT-4o mini
To use synthetic division, we first need to express the divisor in the appropriate form. The divisor given is \( 2x + 1 \). To set up synthetic division, we usually rewrite this as \( 0 = 2x + 1 \) and solve for \( x \):
\[
2x + 1 = 0 \\
2x = -1 \\
x = -\frac{1}{2}
\]
The value we will use in synthetic division is \( -\frac{1}{2} \), which is the root of the divisor \( 2x + 1 \).
Now, we can also note that synthetic division itself will involve coefficients of the dividend \( x^4 + x^3 - 2x^2 + 2x - 5 \), which are \( 1, 1, -2, 2, -5 \).
However, to answer your original question directly: the value of the divisor in terms of \( x \) is \( -\frac{1}{2} \), which is what we use in synthetic division. If you meant to find the divisor in terms of its polynomial expression, the divisor remains \( 2x + 1 \).
\[
2x + 1 = 0 \\
2x = -1 \\
x = -\frac{1}{2}
\]
The value we will use in synthetic division is \( -\frac{1}{2} \), which is the root of the divisor \( 2x + 1 \).
Now, we can also note that synthetic division itself will involve coefficients of the dividend \( x^4 + x^3 - 2x^2 + 2x - 5 \), which are \( 1, 1, -2, 2, -5 \).
However, to answer your original question directly: the value of the divisor in terms of \( x \) is \( -\frac{1}{2} \), which is what we use in synthetic division. If you meant to find the divisor in terms of its polynomial expression, the divisor remains \( 2x + 1 \).