Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

x^(3)+7x^(2)-36
The polynomial has zeros at -6 and 2 . If the remaining zero is
z, then what is the value of
-z ?

$x^{3}+7 x^{2}-36$ $\newline$ The polynomial has zeros at $-6$ and $2$ . If the remaining zero is $z$ , then what is the value of $-z$ ?

View step-by-step help

View step-by-step help

Solve polynomial equations

Full solution

Q.

x^{3}+7 x^{2}-36

\newline

The polynomial has zeros at

-6

and

2

. If the remaining zero is

z

, then what is the value of

-z

?

Factor Theorem: We know that the polynomial $x^3 + 7x^2 - 36$ has zeros at $-6$ and $2$ . Since polynomials are continuous and differentiable everywhere, we can use the Factor Theorem to express the polynomial as a product of its factors. The Factor Theorem states that if $r$ is a root of the polynomial, then $(x - r)$ is a factor of the polynomial.
Finding Factors: First, we can write down the factors corresponding to the known zeros $-6$ and $2$ . This gives us $(x + 6)$ and $(x - 2)$ as factors of the polynomial.
Expanding Factors: Now, we can express the polynomial as the product of its factors and the unknown factor $(x - z)$ , where $z$ is the remaining zero. The polynomial can be written as $(x + 6)(x - 2)(x - z)$ .
Coefficient Matching: To find the value of $z$ , we need to expand the known factors and set them equal to the given polynomial $x^3 + 7x^2 - 36$ . Let's expand $(x + 6)(x - 2)$ first. $\newline$ $(x + 6)(x - 2) = x^2 - 2x + 6x - 12$ $\newline$ $= x^2 + 4x - 12$
Equating Constant Terms: Next, we multiply the expanded expression $(x^2 + 4x - 12)$ by $(x - z)$ and set it equal to the given polynomial $x^3 + 7x^2 - 36$ . However, we can also use the fact that the coefficients of the $x^2$ term in the polynomial must match. The coefficient of $x^2$ in the given polynomial is $7$ , and we already have a coefficient of $1$ from $(x - z)$ , so the coefficient from $(x^2 + 4x - 12)$ must be $6$ .
Solving for z: Since we know that the coefficient of $x^2$ must be $7$ , and we have $x^2$ from $(x - z)$ , the remaining coefficient from $(x^2 + 4x - 12)$ must be $6$ . This means that the coefficient of $x^2$ in $(x^2 + 4x - 12)$ is already correct, and we do not need to adjust the factor $(x - z)$ . Therefore, we can conclude that the polynomial $(x + 6)(x - 2)(x - z)$ is equivalent to the given polynomial $7$ $0$ .
Solving for $z$ : Since we know that the coefficient of $x^2$ must be $7$ , and we have $x^2$ from $(x - z)$ , the remaining coefficient from $(x^2 + 4x - 12)$ must be $6$ . This means that the coefficient of $x^2$ in $(x^2 + 4x - 12)$ is already correct, and we do not need to adjust the factor $(x - z)$ . Therefore, we can conclude that the polynomial $x^2$ $0$ is equivalent to the given polynomial $x^2$ $1$ .Now, we can equate the constant terms from the expanded form of $x^2$ $0$ to the constant term of the given polynomial, which is $x^2$ $3$ . The constant term from $x^2$ $4$ is $x^2$ $5$ , so when we multiply by $(x - z)$ , the constant term must be $x^2$ $3$ . $x^2$ $8$
Solving for z: Since we know that the coefficient of $x^2$ must be $7$ , and we have $x^2$ from $(x - z)$ , the remaining coefficient from $(x^2 + 4x - 12)$ must be $6$ . This means that the coefficient of $x^2$ in $(x^2 + 4x - 12)$ is already correct, and we do not need to adjust the factor $(x - z)$ . Therefore, we can conclude that the polynomial $(x + 6)(x - 2)(x - z)$ is equivalent to the given polynomial $7$ $0$ .Now, we can equate the constant terms from the expanded form of $(x + 6)(x - 2)(x - z)$ to the constant term of the given polynomial, which is $7$ $2$ . The constant term from $7$ $3$ is $7$ $4$ , so when we multiply by $(x - z)$ , the constant term must be $7$ $2$ . $\newline$ $7$ $7$ Solving for z, we divide both sides by $7$ $4$ . $\newline$ $7$ $9$ $\newline$ $x^2$ $0$

More problems from Solve polynomial equations

Question

Find the degree of this polynomial.

\newline

`5b^{10} + 3b^3`

\newline

Posted 3 months ago

Question

\newline

(9a + 5) - (2a + 4)

\newline

Posted 3 months ago

Question

Find the product. Simplify your answer.

\newline

-3v^2(v^2 - 9)

\newline

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

(v - 3)(4v + 1)

\newline

Posted 3 months ago

Question

Find the square. Simplify your answer.

\newline

(3y + 2)^2

\newline

Posted 3 months ago

Question

Divide. If there is a remainder, include it as a simplified fraction.

\newline

(24t^2 + 36t) \div 6t

\newline

Posted 2 months ago

Question

Is the function

q(x) = x^6 - 9

even, odd, or neither?

\newline

\newline

\text{[[even][odd][neither]]}

Posted 3 months ago

Question

Find the product. Simplify your answer.

\newline

-3q^2(-3q^2 + q)

\newline

Posted 3 months ago

Question

Find the product. Simplify your answer.

\newline

(r + 3)(4r + 2)

\newline

Posted 3 months ago

Question

Find the roots of the factored polynomial.

\newline

(x + 7)(x + 4)

\newline

Write your answer as a list of values separated by commas.

\newline

x =

Posted 3 months ago