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

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

3 0

2 years ago

What does the magnitude of a vector represent? a. the direction of the vector c. the length of the vector b. the difference betw

FinnZ [79.3K]

The magnitude of a vector is the length of the vector. It is also known as the modulus of the vector. For any vector of the form <a,b>, the length is equal to sqrt(a^2+b^2). Note how this follows from the pythagorean theorem.

5 0

3 years ago

Read 2 more answers

This is the graph of y = sin(x). Does anyone know the second part?

Alborosie

Answer:

The resulting graph is $y = 5\cdot \sin x$ .

Step-by-step explanation:

The resulting function is of the form:

$y = A\cdot \sin x + k$

Where:

$x$ - Independent variable, dimensionless.

$y$ - Dependent variable, dimensionless.

$A$ - Amplitude, dimensionless.

$k$ - Midpoint value, dimensionless.

The sine function is bounded, between -1 and 1, and must be multiplied by a stretch factor. That is: $A > 0$ . According to the graph, the function is bounded between 5 ( $y_{max}$ ) and -5 ( $y_{min}$ ), and the midpoint value ( $k$ ) is 0. The amplitude is determined by the following calculation:

$A = \frac{y_{max}-y_{min}}{2}$

If $y_{min} = -5$ and $y_{max} = 5$ , then:

$A = 5$

The resulting graph is $y = 5\cdot \sin x$ .

6 0

3 years ago