Understanding the Given Factor

Oct 13, 2025 by miodeya.com 26 views

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Introduction

In algebra, finding the zeros of a polynomial is a crucial concept that helps us understand the behavior of the function. A zero of a polynomial is a value of x that makes the polynomial equal to zero. In this article, we will focus on finding the zeros of a cubic polynomial, specifically the polynomial $f(x) = x^3 - 2x^2 - 14x + 40$ . We will use the given factor $(x + 4)$ to help us find the zeros of the polynomial.

Understanding the Given Factor

The given factor $(x + 4)$ is a binomial that can be factored out of the polynomial $f(x) = x^3 - 2x^2 - 14x + 40$ . This means that we can write the polynomial as a product of the given factor and a quadratic polynomial. To find the quadratic polynomial, we can use polynomial division or synthetic division.

Dividing the Polynomial

To divide the polynomial $f(x) = x^3 - 2x^2 - 14x + 40$ by the given factor $(x + 4)$ , we can use synthetic division. The steps for synthetic division are as follows:

Write the coefficients of the polynomial in a row, with the constant term last.
Write the root of the given factor, which is -4, below the row of coefficients.
Multiply the root by the first coefficient and write the result below the second coefficient.
Add the numbers in the second column and write the result below the third coefficient.
Multiply the root by the result in the second column and write the result below the fourth coefficient.
Add the numbers in the third column and write the result below the fifth coefficient.

Using synthetic division, we get:

| 1 | -2 | -14 | 40 | | -4 | | | | | | -8 | -4 | | | | | -12 | 40 | | | | | 0 |

Finding the Quadratic Polynomial

From the synthetic division, we can see that the remainder is 0, which means that the given factor $(x + 4)$ is a factor of the polynomial $f(x) = x^3 - 2x^2 - 14x + 40$ . The quotient of the division is the quadratic polynomial $x^2 - 6x + 10$ .

Factoring the Quadratic Polynomial

The quadratic polynomial $x^2 - 6x + 10$ can be factored as $(x - 5)(x - 2)$ . Therefore, the polynomial $f(x) = x^3 - 2x^2 - 14x + 40$ can be written as $(x + 4)(x - 5)(x - 2)$ .

Finding the Zeros of the Polynomial

To find the zeros of the polynomial, we need to set each factor equal to zero and solve for x. The factors are $(x + 4)$ , $(x - 5)$ , and $(x - 2)$ . Setting each factor equal to zero, we get:

$(x + 4) = 0 \Rightarrow x = -4$
$(x - 5) = 0 \Rightarrow x = 5$
$(x - 2) = 0 \Rightarrow x = 2$

Conclusion

In conclusion, the zeros of the polynomial $f(x) = x^3 - 2x^2 - 14x + 40$ are $-4, 5, 2$ . These zeros can be found by using the given factor $(x + 4)$ and factoring the polynomial into a product of three binomials.

Final Answer

The final answer is: $\boxed{-4, 5, 2}$

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Q: What is a zero of a polynomial?

A: A zero of a polynomial is a value of x that makes the polynomial equal to zero. In other words, if we substitute the zero into the polynomial, the result will be zero.

Q: How do I find the zeros of a polynomial?

A: To find the zeros of a polynomial, we need to set the polynomial equal to zero and solve for x. We can use various methods such as factoring, synthetic division, or the quadratic formula to find the zeros.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial by a binomial. It is a shortcut for long division and can be used to find the quotient and remainder of the division.

Q: How do I use synthetic division?

A: To use synthetic division, we need to write the coefficients of the polynomial in a row, with the constant term last. We then write the root of the binomial below the row of coefficients. We multiply the root by the first coefficient and write the result below the second coefficient. We add the numbers in the second column and write the result below the third coefficient. We continue this process until we have added all the numbers in the column.

Q: What is the difference between a zero and a root of a polynomial?

A: A zero and a root of a polynomial are the same thing. They are values of x that make the polynomial equal to zero.

Q: Can a polynomial have more than one zero?

A: Yes, a polynomial can have more than one zero. In fact, a polynomial can have as many zeros as its degree. For example, a quadratic polynomial can have two zeros, and a cubic polynomial can have three zeros.

Q: How do I know if a polynomial has a zero at a certain value of x?

A: To determine if a polynomial has a zero at a certain value of x, we can substitute the value of x into the polynomial and see if the result is zero. If the result is zero, then the polynomial has a zero at that value of x.

Q: Can a polynomial have a zero that is not a real number?

A: Yes, a polynomial can have a zero that is not a real number. In fact, a polynomial can have complex zeros, which are zeros that are not real numbers.

Q: How do I find the complex zeros of a polynomial?

A: To find the complex zeros of a polynomial, we can use various methods such as factoring, synthetic division, or the quadratic formula. We can also use numerical methods such as the Newton-Raphson method to find the complex zeros.

Q: What is the significance of finding the zeros of a polynomial?

A: Finding the zeros of a polynomial is important because it helps us understand the behavior of the function. The zeros of a polynomial can tell us about the maximum and minimum values of the function, as well as the intervals where the function is increasing or decreasing.

Q: Can a polynomial have a zero that is not a rational number?

A: Yes, a polynomial can have a zero that is not a rational number. In fact, a polynomial can have irrational zeros, which are zeros that are not rational numbers.

Q: How do I find the irrational zeros of a polynomial?

A: To find the irrational zeros of a polynomial, we can use various methods such as factoring, synthetic division, or the quadratic formula. We can also use numerical methods such as the Newton-Raphson method to find the irrational zeros.

Conclusion

In conclusion, finding the zeros of a polynomial is an important concept in algebra that helps us understand the behavior of the function. We can use various methods such as factoring, synthetic division, or the quadratic formula to find the zeros of a polynomial. We can also use numerical methods such as the Newton-Raphson method to find the complex and irrational zeros of a polynomial.