Question

according to the rational root theorem, the numbers below are some of the potential roots of (f(x) = 10x^4 + 23x^3 - 45x + 21). select all that are actual roots.

(-\frac{7}{2})
(-\frac{3}{5})
(\frac{3}{5})
(\frac{3}{2})
(\frac{7}{10})

Explanation:

🆕 New Concept Discovered: Rational Root Theorem
Finding possible rational solutions of a polynomial.

Step 1: Identify the coefficients

The given polynomial is:

$$ f(x) = 10x^4 + 29x^3 - 41x^2 + 21 $$

Identify the constant term \( p \) and the leading coefficient \( q \):

• Constant term \( p = 21 \)
• Leading coefficient \( q = 10 \)

Step 2: Find the factors of \( p \) and \( q \)

Find all integer factors of \( p \) and \( q \):

• Factors of \( p = 21 \): \( \pm 1, \pm 3, \pm 7, \pm 21 \)
• Factors of \( q = 10 \): \( \pm 1, \pm 2, \pm 5, \pm 10 \)

Step 3: List the possible rational roots

According to the Rational Root Theorem, any rational root must be of the form \( \pm \frac{p}{q} \):

$$ \frac{p}{q} = \frac{\text{Factors of } 21}{\text{Factors of } 10} $$

Forming the possible fractions:

• Using denominator \( 1 \): \( \pm 1, \pm 3, \pm 7, \pm 21 \)
• Using denominator \( 2 \): \( \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{7}{2}, \pm \frac{21}{2} \)
• Using denominator \( 5 \): \( \pm \frac{1}{5}, \pm \frac{3}{5}, \pm \frac{7}{5}, \pm \frac{21}{5} \)
• Using denominator \( 10 \): \( \pm \frac{1}{10}, \pm \frac{3}{10}, \pm \frac{7}{10}, \pm \frac{21}{10} \)

Step 4: Match with the given options

Looking at the options shown in the image:

1. \( \frac{3}{5} \) (This is a possible root since \( 3 \) is a factor of \( 21 \) and \( 5 \) is a factor of \( 10 \))
2. \( \frac{7}{10} \) (This is a possible root since \( 7 \) is a factor of \( 21 \) and \( 10 \) is a factor of \( 10 \))
3. \( \frac{3}{2} \) (This is a possible root since \( 3 \) is a factor of \( 21 \) and \( 2 \) is a factor of \( 10 \))
4. \( 3 \) (This is a possible root since \( 3 \) is a factor of \( 21 \) and \( 1 \) is a factor of \( 10 \))
5. \( 7 \) (This is a possible root since \( 7 \) is a factor of \( 21 \) and \( 1 \) is a factor of \( 10 \))

All of the listed options are possible rational roots of the polynomial.

Answer:

All of the options are possible rational roots:

• \( \frac{3}{5} \)
• \( \frac{7}{10} \)
• \( \frac{3}{2} \)
• \( 3 \)
• \( 7 \)