$\text{According to the factor theorem, if f(b) = 0, then x-b is a factor of f(x).}\\\\\text{Given that,}\\\\f(x) = 2x^3 +5x^2 +4x +1 \\\\f\left(-\dfrac 12 \right) = 2\left( - \dfrac 12 \right)^3 +5 \left( - \dfrac 12 \right)^2 +4 \left( - \dfrac 12 \right) +1 \\\\\\~~~~~~~~~~~~=-2 \cdot \dfrac 18 + 5 \cdot \dfrac 14 -2 +1 \\\\\\~~~~~~~~~~~=\dfrac 54 -\dfrac14 -1\\\\\\~~~~~~~~~~~=\dfrac 44 -1 \\\\\\~~~~~~~~~~~=1-1\\\\\\~~~~~~~~~~~=0\\\\\text{So,}~ 2x +1 ~ \text{is a factor of f(x).}$

$\text{Now,}\\\\f(x) = 2x^3 +5x^2 +4x +1 \\\\~~~~~~=2x^3+x^2 +2x^2 +2x +2x+1\\\\~~~~~~=x^2(2x+1) +2x(2x+1) + (2x+1)\\\\~~~~~~=(2x+1)(x^2 +2x +1)\\\\~~~~~~=(2x+1)(x+1)^2$

3 0

2 years ago

Suppose a circle has a radius of 13 inches. How far would a 24 inch chord be from the center of the circle? (Hint: Draw a diagra

lys-0071 [83]

Answer:

Step-by-step explanation:

Directions

Draw a circle
Dear a chord with a length of 24 inside the circle. You just have to label it as 24
Draw a radius that is perpendicular and a bisector through the chord
Draw a radius that is from the center of the circle to one end of the chord.
Label where the perpendicular radius to the chord intersect. Call it E.
You should get something that looks like the diagram below. The only thing you have to do is put in the point E which is the midpoint of CB.

Givens

AC = 13 inches Given

CB = 24 inches Given

CE = 12 inches Construction and property of a midpoint.

So what we have now is a right triangle (ACE) with the right angle at E.

What we seek is AE

Formula

AC^2 = CE^2 + AE^2

13^2 = 12^2 + AE^2

169 = 144 + AE^2 Subtract 144 from both sides.

169 - 144 = 144-144 + AE^2 Combine

25 = AE^2 Take the square root of both sides