Question
If you were to graph the equations y7x4 and y = 7x +3, how many times would the linear equations cross or touch each other? (1 point)
K
<
<
4
Always
Two times
Never
One time
K
<
<
4
Always
Two times
Never
One time
Answers
GPT-4o mini
To determine how many times the equations \( y = 7x + 3 \) (a linear equation) and \( y = 7x^4 \) (a polynomial equation) cross or touch each other, we can set the two equations equal to each other:
\[
7x + 3 = 7x^4
\]
Rearranging this gives us:
\[
7x^4 - 7x - 3 = 0
\]
Dividing the whole equation by 7 simplifies it to:
\[
x^4 - x - \frac{3}{7} = 0
\]
Now, we'll examine how many real solutions this polynomial equation has.
The left-hand side, \( x^4 - x - \frac{3}{7} \), is a quartic polynomial (degree 4). A quartic polynomial can have up to 4 real roots, but it can also have fewer.
- As \( x \to -\infty \), \( x^4 \) dominates, and the expression tends to \( +\infty \).
- As \( x \to +\infty \), again \( x^4 \) dominates, and the expression tends to \( +\infty \).
- Evaluating the function at \( x = 0 \):
\[
f(0) = 0^4 - 0 - \frac{3}{7} = -\frac{3}{7} < 0
\]
Since \( f(x) \) goes from negative at \( x = 0 \) to positive as \( x \) approaches \( \pm\infty \), there must be at least two changes in sign indicating that there are at least two real roots.
Additionally, since \( x^4 \) grows faster than the linear term, and given the shape of the graph of polynomial functions, it suggests that there could be a total of four real roots, but let's analyze this further using calculus or graphing tools if needed.
However, without visual graphs or deeper calculus, we can conclude based on the behavior at infinity and the single change at zero:
This quartic polynomial \( x^4 - x - \frac{3}{7} \) must intersect the x-axis at least at two points.
Thus, the final answer is:
**Two times.**
\[
7x + 3 = 7x^4
\]
Rearranging this gives us:
\[
7x^4 - 7x - 3 = 0
\]
Dividing the whole equation by 7 simplifies it to:
\[
x^4 - x - \frac{3}{7} = 0
\]
Now, we'll examine how many real solutions this polynomial equation has.
The left-hand side, \( x^4 - x - \frac{3}{7} \), is a quartic polynomial (degree 4). A quartic polynomial can have up to 4 real roots, but it can also have fewer.
- As \( x \to -\infty \), \( x^4 \) dominates, and the expression tends to \( +\infty \).
- As \( x \to +\infty \), again \( x^4 \) dominates, and the expression tends to \( +\infty \).
- Evaluating the function at \( x = 0 \):
\[
f(0) = 0^4 - 0 - \frac{3}{7} = -\frac{3}{7} < 0
\]
Since \( f(x) \) goes from negative at \( x = 0 \) to positive as \( x \) approaches \( \pm\infty \), there must be at least two changes in sign indicating that there are at least two real roots.
Additionally, since \( x^4 \) grows faster than the linear term, and given the shape of the graph of polynomial functions, it suggests that there could be a total of four real roots, but let's analyze this further using calculus or graphing tools if needed.
However, without visual graphs or deeper calculus, we can conclude based on the behavior at infinity and the single change at zero:
This quartic polynomial \( x^4 - x - \frac{3}{7} \) must intersect the x-axis at least at two points.
Thus, the final answer is:
**Two times.**