Factor $f(x)$ Into Linear Factors Given That $k$ Is A Zero Of $ F ( X ) F(x) F ( X ) [/tex].$f(x)=4x 3+11x 2-75x+18 ; K=3$$f(x)=$(Factor Completely.)

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Introduction

In algebra, factoring polynomials is a crucial skill that helps us simplify complex expressions and solve equations. When we are given a polynomial function $f(x)$ and a zero $k$, we can use this information to factor $f(x)$ into linear factors. In this article, we will explore how to factor a polynomial given a zero, using the given polynomial function $f(x) = 4x^3 + 11x^2 - 75x + 18$ and the zero $k = 3$.

Understanding the Problem

To factor a polynomial given a zero, we need to understand the concept of a zero. A zero of a polynomial function $f(x)$ is a value of $x$ that makes $f(x)$ equal to zero. In other words, if $k$ is a zero of $f(x)$, then $f(k) = 0$. Given that $k = 3$ is a zero of $f(x)$, we can substitute $x = 3$ into the polynomial function to get:

f(3)=4(3)3+11(3)2−75(3)+18f(3) = 4(3)^3 + 11(3)^2 - 75(3) + 18

Simplifying the Expression

Now, let's simplify the expression $f(3)$:

f(3)=4(27)+11(9)−75(3)+18f(3) = 4(27) + 11(9) - 75(3) + 18

f(3)=108+99−225+18f(3) = 108 + 99 - 225 + 18

f(3)=0f(3) = 0

Since $f(3) = 0$, we know that $x - 3$ is a factor of $f(x)$. This means that we can write $f(x)$ as:

f(x)=(x−3)⋅g(x)f(x) = (x - 3) \cdot g(x)

where $g(x)$ is a polynomial function.

Finding the Other Factors

To find the other factors of $f(x)$, we need to divide $f(x)$ by $x - 3$. This can be done using polynomial long division or synthetic division. Let's use polynomial long division:

4x2+5x−6x−3\encloselongdiv4x3+11x2−75x+184x3−12x2‾23x2−75x23x2−69x‾6x+186x−18‾36\begin{array}{r} 4x^2 + 5x - 6 \\ x - 3 \enclose{longdiv}{4x^3 + 11x^2 - 75x + 18} \\ \underline{4x^3 - 12x^2} \\ 23x^2 - 75x \\ \underline{23x^2 - 69x} \\ 6x + 18 \\ \underline{6x - 18} \\ 36 \end{array}

Factoring the Polynomial

From the polynomial long division, we can see that the quotient is $4x^2 + 5x - 6$. This means that we can write $f(x)$ as:

f(x)=(x−3)(4x2+5x−6)f(x) = (x - 3)(4x^2 + 5x - 6)

Factoring the Quadratic

Now, we need to factor the quadratic expression $4x^2 + 5x - 6$. We can do this by finding two numbers whose product is $-24$ (the product of the coefficient of $x^2$ and the constant term) and whose sum is $5$ (the coefficient of $x$). These numbers are $8$ and $-3$, so we can write:

4x2+5x−6=(2x+3)(2x−2)4x^2 + 5x - 6 = (2x + 3)(2x - 2)

Factoring the Polynomial Completely

Now, we can factor the polynomial completely:

f(x)=(x−3)(2x+3)(2x−2)f(x) = (x - 3)(2x + 3)(2x - 2)

Conclusion

In this article, we learned how to factor a polynomial given a zero. We used the given polynomial function $f(x) = 4x^3 + 11x^2 - 75x + 18$ and the zero $k = 3$ to factor $f(x)$ into linear factors. We simplified the expression $f(3)$, found the other factors of $f(x)$ using polynomial long division, and factored the quadratic expression $4x^2 + 5x - 6$. Finally, we factored the polynomial completely to get:

f(x)=(x−3)(2x+3)(2x−2)f(x) = (x - 3)(2x + 3)(2x - 2)

Example Problems

Problem 1

Factor the polynomial $f(x) = x^3 + 6x^2 - 11x - 12$ given that $k = 2$ is a zero of $f(x)$.

Solution

To factor the polynomial, we need to substitute $x = 2$ into the polynomial function to get:

f(2)=(2)3+6(2)2−11(2)−12f(2) = (2)^3 + 6(2)^2 - 11(2) - 12

f(2)=8+24−22−12f(2) = 8 + 24 - 22 - 12

f(2)=0f(2) = 0

Since $f(2) = 0$, we know that $x - 2$ is a factor of $f(x)$. This means that we can write $f(x)$ as:

f(x)=(x−2)⋅g(x)f(x) = (x - 2) \cdot g(x)

where $g(x)$ is a polynomial function.

Problem 2

Factor the polynomial $f(x) = 2x^3 - 5x^2 - 13x + 6$ given that $k = 1$ is a zero of $f(x)$.

Solution

To factor the polynomial, we need to substitute $x = 1$ into the polynomial function to get:

f(1)=2(1)3−5(1)2−13(1)+6f(1) = 2(1)^3 - 5(1)^2 - 13(1) + 6

f(1)=2−5−13+6f(1) = 2 - 5 - 13 + 6

f(1)=0f(1) = 0

Since $f(1) = 0$, we know that $x - 1$ is a factor of $f(x)$. This means that we can write $f(x)$ as:

f(x)=(x−1)⋅g(x)f(x) = (x - 1) \cdot g(x)

where $g(x)$ is a polynomial function.

Final Answer

The final answer is: (x−3)(2x+3)(2x−2)\boxed{(x - 3)(2x + 3)(2x - 2)}

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Q: What is a zero of a polynomial function?

A: A zero of a polynomial function $f(x)$ is a value of $x$ that makes $f(x)$ equal to zero. In other words, if $k$ is a zero of $f(x)$, then $f(k) = 0$.

Q: How do I find the other factors of a polynomial given a zero?

A: To find the other factors of a polynomial given a zero, you need to divide the polynomial by the factor that corresponds to the zero. This can be done using polynomial long division or synthetic division.

Q: What is polynomial long division?

A: Polynomial long division is a method for dividing a polynomial by another polynomial. It is similar to long division, but it is used for polynomials instead of numbers.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the product of the coefficient of $x^2$ and the constant term, and whose sum is the coefficient of $x$. These numbers are called the roots of the quadratic expression.

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

A: A root of a quadratic expression is a value of $x$ that makes the expression equal to zero. A zero of a polynomial function is a value of $x$ that makes the function equal to zero.

Q: Can I use synthetic division to factor a polynomial given a zero?

A: Yes, you can use synthetic division to factor a polynomial given a zero. Synthetic division is a method for dividing a polynomial by a linear factor, and it is often faster and easier to use than polynomial long division.

Q: How do I know if a polynomial can be factored completely?

A: A polynomial can be factored completely if it can be written as a product of linear factors. This means that the polynomial must have a zero for each linear factor.

Q: What is the importance of factoring polynomials?

A: Factoring polynomials is an important skill in algebra because it allows us to simplify complex expressions and solve equations. It is also used in many real-world applications, such as physics, engineering, and economics.

Q: Can I use factoring to solve equations?

A: Yes, you can use factoring to solve equations. If you have an equation that involves a polynomial, you can try to factor the polynomial and then use the factored form to solve the equation.

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

  • Not checking if the polynomial can be factored completely
  • Not using the correct method for factoring (such as polynomial long division or synthetic division)
  • Not checking if the factors are linear
  • Not simplifying the expression after factoring

Q: How can I practice factoring polynomials?

A: You can practice factoring polynomials by working through examples and exercises in a textbook or online resource. You can also try to factor polynomials on your own, using the methods and techniques that you have learned.

Q: What are some real-world applications of factoring polynomials?

A: Some real-world applications of factoring polynomials include:

  • Physics: Factoring polynomials is used to solve equations that describe the motion of objects.
  • Engineering: Factoring polynomials is used to design and optimize systems, such as bridges and buildings.
  • Economics: Factoring polynomials is used to model and analyze economic systems.

Q: Can I use factoring to solve systems of equations?

A: Yes, you can use factoring to solve systems of equations. If you have a system of equations that involves polynomials, you can try to factor the polynomials and then use the factored form to solve the system.

Q: What are some common types of polynomials that can be factored?

A: Some common types of polynomials that can be factored include:

  • Quadratic expressions
  • Cubic expressions
  • Quartic expressions
  • Polynomial expressions with rational coefficients

Q: Can I use factoring to solve inequalities?

A: Yes, you can use factoring to solve inequalities. If you have an inequality that involves a polynomial, you can try to factor the polynomial and then use the factored form to solve the inequality.

Q: What are some common mistakes to avoid when factoring inequalities?

A: Some common mistakes to avoid when factoring inequalities include:

  • Not checking if the polynomial can be factored completely
  • Not using the correct method for factoring (such as polynomial long division or synthetic division)
  • Not checking if the factors are linear
  • Not simplifying the expression after factoring

Q: How can I practice factoring inequalities?

A: You can practice factoring inequalities by working through examples and exercises in a textbook or online resource. You can also try to factor inequalities on your own, using the methods and techniques that you have learned.

Q: What are some real-world applications of factoring inequalities?

A: Some real-world applications of factoring inequalities include:

  • Physics: Factoring inequalities is used to solve equations that describe the motion of objects.
  • Engineering: Factoring inequalities is used to design and optimize systems, such as bridges and buildings.
  • Economics: Factoring inequalities is used to model and analyze economic systems.

Q: Can I use factoring to solve systems of inequalities?

A: Yes, you can use factoring to solve systems of inequalities. If you have a system of inequalities that involves polynomials, you can try to factor the polynomials and then use the factored form to solve the system.

Q: What are some common types of inequalities that can be factored?

A: Some common types of inequalities that can be factored include:

  • Linear inequalities
  • Quadratic inequalities
  • Cubic inequalities
  • Polynomial inequalities with rational coefficients

Q: Can I use factoring to solve systems of linear equations?

A: Yes, you can use factoring to solve systems of linear equations. If you have a system of linear equations that involves polynomials, you can try to factor the polynomials and then use the factored form to solve the system.

Q: What are some common types of systems of linear equations that can be factored?

A: Some common types of systems of linear equations that can be factored include:

  • Systems of two linear equations
  • Systems of three linear equations
  • Systems of four linear equations
  • Systems of linear equations with rational coefficients

Q: Can I use factoring to solve systems of quadratic equations?

A: Yes, you can use factoring to solve systems of quadratic equations. If you have a system of quadratic equations that involves polynomials, you can try to factor the polynomials and then use the factored form to solve the system.

Q: What are some common types of systems of quadratic equations that can be factored?

A: Some common types of systems of quadratic equations that can be factored include:

  • Systems of two quadratic equations
  • Systems of three quadratic equations
  • Systems of four quadratic equations
  • Systems of quadratic equations with rational coefficients

Q: Can I use factoring to solve systems of cubic equations?

A: Yes, you can use factoring to solve systems of cubic equations. If you have a system of cubic equations that involves polynomials, you can try to factor the polynomials and then use the factored form to solve the system.

Q: What are some common types of systems of cubic equations that can be factored?

A: Some common types of systems of cubic equations that can be factored include:

  • Systems of two cubic equations
  • Systems of three cubic equations
  • Systems of four cubic equations
  • Systems of cubic equations with rational coefficients

Q: Can I use factoring to solve systems of quartic equations?

A: Yes, you can use factoring to solve systems of quartic equations. If you have a system of quartic equations that involves polynomials, you can try to factor the polynomials and then use the factored form to solve the system.

Q: What are some common types of systems of quartic equations that can be factored?

A: Some common types of systems of quartic equations that can be factored include:

  • Systems of two quartic equations
  • Systems of three quartic equations
  • Systems of four quartic equations
  • Systems of quartic equations with rational coefficients

Q: Can I use factoring to solve systems of polynomial equations with rational coefficients?

A: Yes, you can use factoring to solve systems of polynomial equations with rational coefficients. If you have a system of polynomial equations that involves polynomials with rational coefficients, you can try to factor the polynomials and then use the factored form to solve the system.

Q: What are some common types of systems of polynomial equations with rational coefficients that can be factored?

A: Some common types of systems of polynomial equations with rational coefficients that can be factored include:

  • Systems of two polynomial equations with rational coefficients
  • Systems of three polynomial equations with rational coefficients
  • Systems of four polynomial equations with rational coefficients
  • Systems of polynomial equations with rational