Answer:
- there are 4 complex solutions
- 3 real zeros and 2 complex zeros
Step-by-step explanation:
1. Descarte's rule of signs tells you there are 0 positive real roots and 0 or 2 negative real roots. (for positive x, signs are ++++ so have no changes; for negative x, signs are ++-+, so have 2 changes.) A graph shows no real roots.
2. There are 3 sign changes in the given polynomial, so 3 or 1 positive real roots. When the sign of x is changed, there are 2 sign changes, so 0 or 2 negative real roots. A graph shows 2 negative and one positive real root (for a total of 3), so the remaining 2 roots are complex.
1.5/4 is shaded, since there is one block shaded, and half of another block (1.5)
let's first off take a peek at those values.
let's say the point with those coordinates is point C, so C is 3/10 of the way from A to B.
meaning, we take the segment AB and cut it in 10 equal pieces, AC takes 3 pieces, and CB takes 7 pieces, namely AC and CB are at a 3:7 ratio.
![\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ A(-4,-8)\qquad B(11,7)\qquad \qquad \stackrel{\textit{ratio from A to B}}{3:7} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{3}{7}\implies \cfrac{A}{B} = \cfrac{3}{7}\implies 7A=3B\implies 7(-4,-8)=3(11,7)\\\\[-0.35em] ~\dotfill\\\\ C=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em] ~\dotfill](https://tex.z-dn.net/?f=%5Cbf%20~~~~~~~~~~~~%5Ctextit%7Binternal%20division%20of%20a%20line%20segment%7D%0A%5C%5C%5C%5C%5C%5C%0AA%28-4%2C-8%29%5Cqquad%20B%2811%2C7%29%5Cqquad%0A%5Cqquad%20%5Cstackrel%7B%5Ctextit%7Bratio%20from%20A%20to%20B%7D%7D%7B3%3A7%7D%0A%5C%5C%5C%5C%5C%5C%0A%5Ccfrac%7BA%5Cunderline%7BC%7D%7D%7B%5Cunderline%7BC%7D%20B%7D%20%3D%20%5Ccfrac%7B3%7D%7B7%7D%5Cimplies%20%5Ccfrac%7BA%7D%7BB%7D%20%3D%20%5Ccfrac%7B3%7D%7B7%7D%5Cimplies%207A%3D3B%5Cimplies%207%28-4%2C-8%29%3D3%2811%2C7%29%5C%5C%5C%5C%5B-0.35em%5D%0A~%5Cdotfill%5C%5C%5C%5C%0AC%3D%5Cleft%28%5Cfrac%7B%5Ctextit%7Bsum%20of%20%22x%22%20values%7D%7D%7B%5Ctextit%7Bsum%20of%20ratios%7D%7D%5Cquad%20%2C%5Cquad%20%5Cfrac%7B%5Ctextit%7Bsum%20of%20%22y%22%20values%7D%7D%7B%5Ctextit%7Bsum%20of%20ratios%7D%7D%5Cright%29%5C%5C%5C%5C%5B-0.35em%5D%0A~%5Cdotfill)
![\bf C=\left(\cfrac{(7\cdot -4)+(3\cdot 11)}{3+7}\quad ,\quad \cfrac{(7\cdot -8)+(3\cdot 7)}{3+7}\right) \\\\\\ C=\left( \cfrac{-28+33}{10}~~,~~\cfrac{-56+21}{10} \right)\implies C=\left( \cfrac{5}{10}~~,~~\cfrac{-35}{10} \right) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill C=\left( \frac{1}{2}~,~-\frac{7}{2} \right)~\hfill](https://tex.z-dn.net/?f=%5Cbf%20C%3D%5Cleft%28%5Ccfrac%7B%287%5Ccdot%20-4%29%2B%283%5Ccdot%2011%29%7D%7B3%2B7%7D%5Cquad%20%2C%5Cquad%20%5Ccfrac%7B%287%5Ccdot%20-8%29%2B%283%5Ccdot%207%29%7D%7B3%2B7%7D%5Cright%29%0A%5C%5C%5C%5C%5C%5C%0AC%3D%5Cleft%28%20%5Ccfrac%7B-28%2B33%7D%7B10%7D~~%2C~~%5Ccfrac%7B-56%2B21%7D%7B10%7D%20%5Cright%29%5Cimplies%20C%3D%5Cleft%28%20%5Ccfrac%7B5%7D%7B10%7D~~%2C~~%5Ccfrac%7B-35%7D%7B10%7D%20%5Cright%29%0A%5C%5C%5C%5C%5B-0.35em%5D%0A%5Crule%7B34em%7D%7B0.25pt%7D%5C%5C%5C%5C%0A~%5Chfill%20C%3D%5Cleft%28%20%5Cfrac%7B1%7D%7B2%7D~%2C~-%5Cfrac%7B7%7D%7B2%7D%20%5Cright%29~%5Chfill)
Distribute
x-12x+30=-x-4
Combine like terms
-11x+30=-x-4
Add x to both sides
-10x+30= -4
Subtract 30 from both sides
-10x=-34
Divide both sides by -10
x=3.4
Final answer: x=3.4
Answer:
4
Step-by-step explanation:
A coefficient is a number before the variable. The third term is 4c^3, and the coefficient is 4.
Hope this helps!
