Type The Correct Answer In Each Box. Use Numerals Instead Of Words. If Necessary, Use / For The Fraction Bar(s).The Degree Of The Function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ Is $\square$, And Its $y$-intercept Is

The degree of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ is $\square$, and its $y$-intercept is

Understanding the Degree of a Polynomial Function

The degree of a polynomial function is the highest power or exponent of the variable in the function. In the given function $f(x) = -(x+1)^2(2x-3)(x+2)^2$, we need to find the highest power of $x$ to determine the degree of the function.

Breaking Down the Function

To find the degree of the function, we need to break down the function into its individual terms. The function can be written as:

$f(x) = -(x+1)^2(2x-3)(x+2)^2$

We can start by expanding the squared terms:

$f(x) = -(x^2 + 2x + 1)(2x-3)(x^2 + 4x + 4)$

Now, we need to multiply the three terms together to get the expanded form of the function.

Multiplying the Terms

To multiply the three terms together, we need to multiply each term in the first parentheses by each term in the second parentheses, and then multiply each term in the result by each term in the third parentheses.

$f(x) = -(x^2 + 2x + 1)(2x-3)(x^2 + 4x + 4)$

$f(x) = -(2x^3 - 3x^2 + 4x^2 - 6x + 2x^2 - 3x + x^2 + 4x + 4)$

$f(x) = -(2x^3 + 4x^2 - 3x^2 - 6x - 3x + x^2 + 4x + 4)$

$f(x) = -(2x^3 + x^2 - 9x + 4)$

Finding the Degree of the Function

Now that we have the expanded form of the function, we can find the degree of the function by looking at the highest power of $x$. In this case, the highest power of $x$ is $3$, so the degree of the function is $\boxed{3}$.

Finding the $y$-Intercept

The $y$-intercept of a function is the point where the function intersects the $y$-axis. To find the $y$-intercept, we need to find the value of $f(x)$ when $x = 0$.

$f(0) = -(0+1)^2(2(0)-3)(0+2)^2$

$f(0) = -(1)^2(-3)(4)$

$f(0) = -12$

Therefore, the $y$-intercept of the function is $-12$.

Conclusion

In conclusion, the degree of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ is $\boxed{3}$, and its $y$-intercept is $-12$.

The degree of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ is $\square$, and its $y$-intercept is

Understanding the Degree of a Polynomial Function

The degree of a polynomial function is the highest power or exponent of the variable in the function. In the given function $f(x) = -(x+1)^2(2x-3)(x+2)^2$, we need to find the highest power of $x$ to determine the degree of the function.

Breaking Down the Function

To find the degree of the function, we need to break down the function into its individual terms. The function can be written as:

$f(x) = -(x+1)^2(2x-3)(x+2)^2$

We can start by expanding the squared terms:

$f(x) = -(x^2 + 2x + 1)(2x-3)(x^2 + 4x + 4)$

Now, we need to multiply the three terms together to get the expanded form of the function.

Multiplying the Terms

To multiply the three terms together, we need to multiply each term in the first parentheses by each term in the second parentheses, and then multiply each term in the result by each term in the third parentheses.

$f(x) = -(x^2 + 2x + 1)(2x-3)(x^2 + 4x + 4)$

$f(x) = -(2x^3 - 3x^2 + 4x^2 - 6x + 2x^2 - 3x + x^2 + 4x + 4)$

$f(x) = -(2x^3 + 4x^2 - 3x^2 - 6x - 3x + x^2 + 4x + 4)$

$f(x) = -(2x^3 + x^2 - 9x + 4)$

Finding the Degree of the Function

Now that we have the expanded form of the function, we can find the degree of the function by looking at the highest power of $x$. In this case, the highest power of $x$ is $3$, so the degree of the function is $\boxed{3}$.

Finding the $y$-Intercept

The $y$-intercept of a function is the point where the function intersects the $y$-axis. To find the $y$-intercept, we need to find the value of $f(x)$ when $x = 0$.

$f(0) = -(0+1)^2(2(0)-3)(0+2)^2$

$f(0) = -(1)^2(-3)(4)$

$f(0) = -12$

Therefore, the $y$-intercept of the function is $-12$.

Conclusion

In conclusion, the degree of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ is $\boxed{3}$, and its $y$-intercept is $-12$.

Q&A

Q: What is the degree of a polynomial function?

A: The degree of a polynomial function is the highest power or exponent of the variable in the function.

Q: How do you find the degree of a polynomial function?

A: To find the degree of a polynomial function, you need to look at the highest power of the variable in the function.

Q: What is the $y$-intercept of a function?

A: The $y$-intercept of a function is the point where the function intersects the $y$-axis.

Q: How do you find the $y$-intercept of a function?

A: To find the $y$-intercept of a function, you need to find the value of the function when $x = 0$.

Q: What is the degree of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$?

A: The degree of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ is $\boxed{3}$.

Q: What is the $y$-intercept of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$?

A: The $y$-intercept of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ is $-12$.

Q: How do you multiply three terms together?

A: To multiply three terms together, you need to multiply each term in the first parentheses by each term in the second parentheses, and then multiply each term in the result by each term in the third parentheses.

Q: What is the expanded form of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$?

A: The expanded form of the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ is $-(2x^3 + x^2 - 9x + 4)$.

Q: What is the highest power of $x$ in the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$?

A: The highest power of $x$ in the function $f(x) = -(x+1)^2(2x-3)(x+2)^2$ is $3$.