Find the x-intercepts of the polynomial function k(x)=4x(1−2x)(3x+4)

Question

.(1 point)
Responses

(−43,0)
, (0,0)
, (2,0)
left parenthesis negative Start Fraction 4 over 3 End Fraction comma 0 right parenthesis , left parenthesis 0 comma 0 right parenthesis , left parenthesis 2 comma 0 right parenthesis

(−43,0)
, (0,0)
, (12,0)
left parenthesis negative Start Fraction 4 over 3 End Fraction comma 0 right parenthesis , left parenthesis 0 comma 0 right parenthesis , left parenthesis Start Fraction 1 over 2 End Fraction comma 0 right parenthesis

(−43,0)
, (12,0)
, (4,0)

left parenthesis negative Start Fraction 4 over 3 End Fraction comma 0 right parenthesis , left ​parenthesis Start Fraction 1 over 2 End Fraction comma 0 right parenthesis , left parenthesis 4 comma 0 right parenthesis

(−43,0)
, (2,0)
, (4,0)

Answer 1

To find the x-intercepts of the polynomial function \( k(x) = 4x(1 − 2x)(3x + 4) \), we set the function equal to zero:

\[ 4x(1 - 2x)(3x + 4) = 0 \]

This equation will be zero if any of its factors is zero. We can find the x-intercepts by solving each factor individually:

1. From the first factor: \( 4x = 0 \)

   • This gives: \( x = 0 \)

2. From the second factor: \( 1 - 2x = 0 \)

   • Solving for \( x\): \[ 2x = 1 \implies x = \frac{1}{2} \]

3. From the third factor: \( 3x + 4 = 0 \)

   • Solving for \( x\): \[ 3x = -4 \implies x = -\frac{4}{3} \]

So the x-intercepts of the function \( k(x) \) are:

• \( \left(0, 0\right) \)
• \( \left(-\frac{4}{3}, 0\right) \)
• \( \left(\frac{1}{2}, 0\right) \)

Therefore, the correct response is: \[ \left( -\frac{4}{3}, 0 \right), \left( 0, 0 \right), \left( \frac{1}{2}, 0 \right) \]