Question
Factor completely.
8k4–4k3–10k2+5k
Answers
Answered by
GPT-4o mini
To factor the expression \( 8k^4 - 4k^3 - 10k^2 + 5k \) completely, we can start by factoring out the greatest common factor (GCF) of the terms.
1. Identify the GCF of the coefficients: The GCF of \( 8, -4, -10, \) and \( 5 \) is \( 1 \) (since \( 5 \) is prime and doesn't divide the other coefficients).
2. Identify the GCF of the variable terms: The lowest power of \( k \) that appears in all terms is \( k \).
So the GCF of the entire expression is \( k \).
Now, let's factor out \( k \):
\[
k(8k^3 - 4k^2 - 10k + 5)
\]
Next, we'll focus on factoring the cubic polynomial \( 8k^3 - 4k^2 - 10k + 5 \).
We will use the Rational Root Theorem to find possible rational roots. Possible rational roots include \( \pm1, \pm5, \pm\frac{1}{2}, \pm\frac{5}{2}, \pm\frac{1}{4}, \pm\frac{5}{4} \).
Testing \( k = 1 \):
\[
8(1)^3 - 4(1)^2 - 10(1) + 5 = 8 - 4 - 10 + 5 = -1 \quad (\text{not a root})
\]
Testing \( k = -1 \):
\[
8(-1)^3 - 4(-1)^2 - 10(-1) + 5 = -8 - 4 + 10 + 5 = 3 \quad (\text{not a root})
\]
Testing \( k = \frac{1}{2} \):
\[
8\left(\frac{1}{2}\right)^3 - 4\left(\frac{1}{2}\right)^2 - 10\left(\frac{1}{2}\right) + 5 = 8\left(\frac{1}{8}\right) - 4\left(\frac{1}{4}\right) - 5 + 5 = 1 - 1 - 5 + 5 = 0 \quad (\text{is a root})
\]
So, \( k = \frac{1}{2} \) is a root. We will perform synthetic division on \( 8k^3 - 4k^2 - 10k + 5 \) using \( k - \frac{1}{2} \):
1. Write the coefficients: \( 8, -4, -10, 5 \)
2. Using \( \frac{1}{2} \) in synthetic division:
\[
\begin{array}{r|rrrr}
\frac{1}{2} & 8 & -4 & -10 & 5 \\
& & 4 & 0 & -5 \\
\hline
& 8 & 0 & -10 & 0 \\
\end{array}
\]
The result of the synthetic division is \( 8k^2 - 10 \). So now we can write:
\[
8k^3 - 4k^2 - 10k + 5 = (k - \frac{1}{2})(8k^2 - 10)
\]
Next, we can factor \( 8k^2 - 10 \):
\[
8k^2 - 10 = 2(4k^2 - 5)
\]
Thus, we can express the original expression as:
\[
k(2)(4k^2 - 5)(k - \frac{1}{2}) = 2k(4k^2 - 5)(k - \frac{1}{2})
\]
Finally, we write the complete factorization of the original polynomial:
\[
\boxed{2k(4k^2 - 5)(k - \frac{1}{2})}
\]
1. Identify the GCF of the coefficients: The GCF of \( 8, -4, -10, \) and \( 5 \) is \( 1 \) (since \( 5 \) is prime and doesn't divide the other coefficients).
2. Identify the GCF of the variable terms: The lowest power of \( k \) that appears in all terms is \( k \).
So the GCF of the entire expression is \( k \).
Now, let's factor out \( k \):
\[
k(8k^3 - 4k^2 - 10k + 5)
\]
Next, we'll focus on factoring the cubic polynomial \( 8k^3 - 4k^2 - 10k + 5 \).
We will use the Rational Root Theorem to find possible rational roots. Possible rational roots include \( \pm1, \pm5, \pm\frac{1}{2}, \pm\frac{5}{2}, \pm\frac{1}{4}, \pm\frac{5}{4} \).
Testing \( k = 1 \):
\[
8(1)^3 - 4(1)^2 - 10(1) + 5 = 8 - 4 - 10 + 5 = -1 \quad (\text{not a root})
\]
Testing \( k = -1 \):
\[
8(-1)^3 - 4(-1)^2 - 10(-1) + 5 = -8 - 4 + 10 + 5 = 3 \quad (\text{not a root})
\]
Testing \( k = \frac{1}{2} \):
\[
8\left(\frac{1}{2}\right)^3 - 4\left(\frac{1}{2}\right)^2 - 10\left(\frac{1}{2}\right) + 5 = 8\left(\frac{1}{8}\right) - 4\left(\frac{1}{4}\right) - 5 + 5 = 1 - 1 - 5 + 5 = 0 \quad (\text{is a root})
\]
So, \( k = \frac{1}{2} \) is a root. We will perform synthetic division on \( 8k^3 - 4k^2 - 10k + 5 \) using \( k - \frac{1}{2} \):
1. Write the coefficients: \( 8, -4, -10, 5 \)
2. Using \( \frac{1}{2} \) in synthetic division:
\[
\begin{array}{r|rrrr}
\frac{1}{2} & 8 & -4 & -10 & 5 \\
& & 4 & 0 & -5 \\
\hline
& 8 & 0 & -10 & 0 \\
\end{array}
\]
The result of the synthetic division is \( 8k^2 - 10 \). So now we can write:
\[
8k^3 - 4k^2 - 10k + 5 = (k - \frac{1}{2})(8k^2 - 10)
\]
Next, we can factor \( 8k^2 - 10 \):
\[
8k^2 - 10 = 2(4k^2 - 5)
\]
Thus, we can express the original expression as:
\[
k(2)(4k^2 - 5)(k - \frac{1}{2}) = 2k(4k^2 - 5)(k - \frac{1}{2})
\]
Finally, we write the complete factorization of the original polynomial:
\[
\boxed{2k(4k^2 - 5)(k - \frac{1}{2})}
\]