Answer:
8. B) sometimes true
9. A) always true
10. A) always true
Step-by-step explanation:
8. A polynomial function with real coefficients has real zeros. This is sometimes true because say we have an equation x^2+10. If you graph this, you will notice that there are no points on the x axis, or no zeroes. That means that this equation would have complex roots. x^2+10 is a real number equation without any complex numbers so the answer is sometimes true.
9. A polynomial function that does not intercept the x-axis has complex roots only. a If a line crosses or touches the x (horizontal) axis on a graph then you can say that the point where it touches is a root. If any equation on a graph does not have a point on the x axis then it has complex roots. So the answer is always true.
10. An odd degree polynomial function with real coefficients has at least one real root. a Think about all the odd degree equations you know. y=x^1 (basically y=x), y=x^3, etc. A linear slope intercept form equation has an odd degree of 1 and passes through the x axis only once. If you plug in x^3 on a graphing calculator or desmos.com/calculator you will see that the y range has infinately many solutions. You can mess with this and plug in whacky equations like x^3-42x^2+16.92x-.4^8; no matter what, there will always be atleast 1 real root, so this is always true.
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Answer:
Hence, the correct options are as follow (8) option B (9) option A (10) option A.
Step-by-step explanation:
Given: Statements about Polynomial function
Solving each questions one by one
(8) A polynomial function with real coefficients has real zeros is sometimes true as say we have an equation [tex]x^2+9[/tex]. If you graph this, you will notice that there are no points on the [tex]x-[/tex]axis or no zeroes. That means that this equation would have complex roots. Thats why it is sometimes true.
Therefore, the statement is sometimes true.
(9) A polynomial function that does not intercept the [tex]x-[/tex]axis has complex roots only is always true as If a line crosses or touches the [tex]x-[/tex]axis (horizontal) on a graph then you can say that the point where it touches is a root. If any equation on a graph does not have a point on the [tex]x-[/tex]axis then it has the complex roots.
Therefore, the statement is Always true.
(10) An odd degree polynomial function with real coefficients has at least one real root is always true as consider all the odd degree equations you know. [tex]y=x^1\;(y=x)\;\&\;y=x^3[/tex], etc. A linear slope intercept form equation has an odd degree of 1 and passes through the [tex]x-[/tex]axis only once.
Therefore, the statement is Always true.
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