Question
Use Synthetic Division and the Remainder Theorem to determine if (x+2) is a factor of the polynomial:
f(x) = x^5 − 3x^4 − 9x^3 + 23x^2 + 24x− 36
f(x) = x^5 − 3x^4 − 9x^3 + 23x^2 + 24x− 36
Answers
GPT-4o mini
To determine if \( (x + 2) \) is a factor of the polynomial \( f(x) = x^5 - 3x^4 - 9x^3 + 23x^2 + 24x - 36 \), we can use the Remainder Theorem. According to the theorem, if \( (x - c) \) is a factor of \( f(x) \), then \( f(c) = 0 \).
Since we have \( (x + 2) \), we can set \( c = -2 \). We will evaluate \( f(-2) \).
The polynomial we have is:
\[
f(x) = x^5 - 3x^4 - 9x^3 + 23x^2 + 24x - 36
\]
Now we need to calculate \( f(-2) \):
\[
f(-2) = (-2)^5 - 3(-2)^4 - 9(-2)^3 + 23(-2)^2 + 24(-2) - 36
\]
Calculating each term:
1. \( (-2)^5 = -32 \)
2. \( -3(-2)^4 = -3(16) = -48 \)
3. \( -9(-2)^3 = -9(-8) = 72 \)
4. \( 23(-2)^2 = 23(4) = 92 \)
5. \( 24(-2) = -48 \)
6. The constant term is \( -36 \)
Now substituting these values into the equation:
\[
f(-2) = -32 - 48 + 72 + 92 - 48 - 36
\]
Now, perform the addition and subtraction step by step:
\[
f(-2) = -32 - 48 = -80
\]
\[
f(-2) = -80 + 72 = -8
\]
\[
f(-2) = -8 + 92 = 84
\]
\[
f(-2) = 84 - 48 = 36
\]
\[
f(-2) = 36 - 36 = 0
\]
Since \( f(-2) = 0 \), this means that \( (x + 2) \) is indeed a factor of the polynomial \( f(x) \).
Thus, the conclusion is:
\[
\text{Yes, } (x + 2) \text{ is a factor of } f(x).
\]
Since we have \( (x + 2) \), we can set \( c = -2 \). We will evaluate \( f(-2) \).
The polynomial we have is:
\[
f(x) = x^5 - 3x^4 - 9x^3 + 23x^2 + 24x - 36
\]
Now we need to calculate \( f(-2) \):
\[
f(-2) = (-2)^5 - 3(-2)^4 - 9(-2)^3 + 23(-2)^2 + 24(-2) - 36
\]
Calculating each term:
1. \( (-2)^5 = -32 \)
2. \( -3(-2)^4 = -3(16) = -48 \)
3. \( -9(-2)^3 = -9(-8) = 72 \)
4. \( 23(-2)^2 = 23(4) = 92 \)
5. \( 24(-2) = -48 \)
6. The constant term is \( -36 \)
Now substituting these values into the equation:
\[
f(-2) = -32 - 48 + 72 + 92 - 48 - 36
\]
Now, perform the addition and subtraction step by step:
\[
f(-2) = -32 - 48 = -80
\]
\[
f(-2) = -80 + 72 = -8
\]
\[
f(-2) = -8 + 92 = 84
\]
\[
f(-2) = 84 - 48 = 36
\]
\[
f(-2) = 36 - 36 = 0
\]
Since \( f(-2) = 0 \), this means that \( (x + 2) \) is indeed a factor of the polynomial \( f(x) \).
Thus, the conclusion is:
\[
\text{Yes, } (x + 2) \text{ is a factor of } f(x).
\]
32
could you give the explanation, but without an excessive use of parenthesis and slashes?
GPT-4o mini
Sure! To determine whether \( (x + 2) \) is a factor of the polynomial \( f(x) = x^5 - 3x^4 - 9x^3 + 23x^2 + 24x - 36 \), we can use the Remainder Theorem. This theorem states that if \( (x - c) \) is a factor of \( f(x) \), then the value of \( f(c) \) will be zero.
Since we are checking for the factor \( (x + 2) \), we can set \( c = -2 \) and calculate \( f(-2) \).
Now we substitute \(-2\) into the polynomial:
1. Calculate \( (-2)^5 \), which equals \(-32\).
2. Calculate \(-3 \times (-2)^4\). Since \((-2)^4\) equals \(16\), this gives us \(-3 \times 16 = -48\).
3. Calculate \(-9 \times (-2)^3\). Here, \((-2)^3\) equals \(-8\), so this becomes \(72\).
4. Calculate \(23 \times (-2)^2\). Since \((-2)^2\) equals \(4\), this is \(92\).
5. Calculate \(24 \times (-2)\), which gives us \(-48\).
6. The constant term is \(-36\).
Now, we can combine all these results:
1. Start with \(-32\).
2. Add \(-48\), resulting in \(-80\).
3. Add \(72\), giving \(-8\).
4. Add \(92\) to get \(84\).
5. Subtract \(48\) to arrive at \(36\).
6. Finally, subtract \(36\) to get \(0\).
Since \( f(-2) = 0 \), it confirms that \( (x + 2) \) is a factor of the polynomial \( f(x) \).
In summary, yes, \( (x + 2) \) is a factor of \( f(x) \).
Since we are checking for the factor \( (x + 2) \), we can set \( c = -2 \) and calculate \( f(-2) \).
Now we substitute \(-2\) into the polynomial:
1. Calculate \( (-2)^5 \), which equals \(-32\).
2. Calculate \(-3 \times (-2)^4\). Since \((-2)^4\) equals \(16\), this gives us \(-3 \times 16 = -48\).
3. Calculate \(-9 \times (-2)^3\). Here, \((-2)^3\) equals \(-8\), so this becomes \(72\).
4. Calculate \(23 \times (-2)^2\). Since \((-2)^2\) equals \(4\), this is \(92\).
5. Calculate \(24 \times (-2)\), which gives us \(-48\).
6. The constant term is \(-36\).
Now, we can combine all these results:
1. Start with \(-32\).
2. Add \(-48\), resulting in \(-80\).
3. Add \(72\), giving \(-8\).
4. Add \(92\) to get \(84\).
5. Subtract \(48\) to arrive at \(36\).
6. Finally, subtract \(36\) to get \(0\).
Since \( f(-2) = 0 \), it confirms that \( (x + 2) \) is a factor of the polynomial \( f(x) \).
In summary, yes, \( (x + 2) \) is a factor of \( f(x) \).