Applying the Remainder Theorem to the Cubic Function

Oct 13, 2025 by windailys.com 111 views

Introduction

When given a polynomial function and one of its roots, we can use the Remainder Theorem to find the other factors of the function. In this article, we will explore how to use the Remainder Theorem to factor a cubic function, given one of its roots. We will apply this method to the cubic function $$f(x)=x^3-4x2-20x+48$$, where one of its roots is $$x=6$$.

The Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that relates the remainder of a polynomial function when divided by a linear factor to the value of the function at a specific point. If a polynomial function $$f(x)$$ is divided by $$x-a$$, then the remainder is equal to $$f(a)$$. This theorem can be used to find the remainder of a polynomial function when divided by a linear factor, and it can also be used to find the factors of a polynomial function when one of its roots is known.

Applying the Remainder Theorem to the Cubic Function

Given the cubic function $$f(x)=x^3-4x2-20x+48$$ and one of its roots $$x=6$$, we can use the Remainder Theorem to find the other factors of the function. According to the Remainder Theorem, if $$x=6$$ is a root of the function, then $$f(6)=0$$. We can substitute $$x=6$$ into the function to verify this:

f(6)=(6)^3-4(6)^2-20(6)+48

f(6)=216-144-120+48

f(6)=0

This confirms that $$x=6$$ is a root of the function.

Factoring the Cubic Function

Now that we have verified that $$x=6$$ is a root of the function, we can use the Remainder Theorem to find the other factors of the function. We know that the function can be written in the form:

f(x)=(x-6)(x^2+ax+b)

where $$a$$ and $$b$$ are constants to be determined.

Using Synthetic Division to Find the Other Factors

To find the other factors of the function, we can use synthetic division to divide the function by the linear factor $$(x-6)$$. Synthetic division is a method of dividing a polynomial function by a linear factor, and it can be used to find the quotient and remainder of the division.

Performing Synthetic Division

To perform synthetic division, we need to write the coefficients of the function in a row, with the constant term on the right-hand side. We then bring down the first coefficient, multiply it by the root, and add the result to the next coefficient. We repeat this process until we have multiplied the last coefficient by the root and added the result to the constant term.

Synthetic Division Table

1	-4	-20	48
6

Performing the Synthetic Division

We start by bringing down the first coefficient, which is 1. We then multiply it by the root, which is 6, and add the result to the next coefficient, which is -4. This gives us:

1 \times 6 = 6

6 + (-4) = 2

We then bring down the next coefficient, which is -20. We multiply it by the root, which is 6, and add the result to the previous result, which is 2. This gives us:

2 + (-20 \times 6) = 2 - 120 = -118

We then bring down the constant term, which is 48. We multiply it by the root, which is 6, and add the result to the previous result, which is -118. This gives us:

-118 + (48 \times 6) = -118 + 288 = 170

Writing the Quotient and Remainder

The quotient of the synthetic division is the polynomial function $$x^2+2x-118$$, and the remainder is 170.

Writing the Factored Form of the Function

We can now write the factored form of the function as:

f(x)=(x-6)(x^2+2x-118)

Factoring the Quadratic Function

To factor the quadratic function $$x^2+2x-118$$, we can use the quadratic formula or factor it by grouping. We will use the quadratic formula to find the roots of the quadratic function.

Quadratic Formula

The quadratic formula is given by:

x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}

where $$a$$, $$b$$, and $$c$$ are the coefficients of the quadratic function.

Applying the Quadratic Formula

We can apply the quadratic formula to the quadratic function $$x^2+2x-118$$ by substituting the values of $$a$$, $$b$$, and $$c$$ into the formula. We have:

a=1

b=2

c=-118

Substituting these values into the quadratic formula, we get:

x=\frac{-2 \pm \sqrt{2^2-4(1)(-118)}}{2(1)}

x=\frac{-2 \pm \sqrt{4+472}}{2}

x=\frac{-2 \pm \sqrt{476}}{2}

x=\frac{-2 \pm 22}{2}

This gives us two possible values for $$x$$:

x_1=\frac{-2 + 22}{2} = 10

x_2=\frac{-2 - 22}{2} = -12

Writing the Factored Form of the Function

We can now write the factored form of the function as:

f(x)=(x-6)(x-10)(x+12)

Conclusion

In this article, we used the Remainder Theorem to find the other factors of the cubic function $$f(x)=x^3-4x2-20x+48$$, given one of its roots $$x=6$$. We applied the Remainder Theorem to find the quotient and remainder of the division of the function by the linear factor $$(x-6)$$. We then used synthetic division to find the other factors of the function. Finally, we factored the quadratic function $$x^2+2x-118$$ using the quadratic formula. The factored form of the function is $$f(x)=(x-6)(x-10)(x+12)$$.

Final Answer

The final answer is: $\boxed{(x-6)(x-10)(x+12)}$

Introduction

In our previous article, we used the Remainder Theorem to find the other factors of the cubic function $$f(x)=x^3-4x2-20x+48$$, given one of its roots $$x=6$$. We applied the Remainder Theorem to find the quotient and remainder of the division of the function by the linear factor $$(x-6)$$, and then used synthetic division to find the other factors of the function. Finally, we factored the quadratic function $$x^2+2x-118$$ using the quadratic formula. In this article, we will answer some common questions related to the topic.

Q&A

Q: What is the Remainder Theorem?

A: The Remainder Theorem is a fundamental concept in algebra that relates the remainder of a polynomial function when divided by a linear factor to the value of the function at a specific point. If a polynomial function $$f(x)$$ is divided by $$x-a$$, then the remainder is equal to $$f(a)$$.

Q: How do I apply the Remainder Theorem to find the other factors of a polynomial function?

A: To apply the Remainder Theorem, you need to know one of the roots of the polynomial function. You can then use the Remainder Theorem to find the quotient and remainder of the division of the function by the linear factor corresponding to the known root. You can then use synthetic division to find the other factors of the function.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial function by a linear factor. It is a shortcut method that allows you to find the quotient and remainder of the division of the function by the linear factor without having to perform long division.

Q: How do I perform synthetic division?

A: To perform synthetic division, you need to write the coefficients of the polynomial function in a row, with the constant term on the right-hand side. You then bring down the first coefficient, multiply it by the root, and add the result to the next coefficient. You repeat this process until you have multiplied the last coefficient by the root and added the result to the constant term.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows you to find the roots of a quadratic function. It is given by:

x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}

where $$a$$, $$b$$, and $$c$$ are the coefficients of the quadratic function.

Q: How do I apply the quadratic formula to find the roots of a quadratic function?

A: To apply the quadratic formula, you need to know the coefficients of the quadratic function. You can then substitute these values into the quadratic formula to find the roots of the function.

Q: What is the factored form of a polynomial function?

A: The factored form of a polynomial function is a way of writing the function as a product of linear factors. Each linear factor corresponds to a root of the function.

Q: How do I find the factored form of a polynomial function?

A: To find the factored form of a polynomial function, you need to know the roots of the function. You can then use the factored form of the function to write it as a product of linear factors.

Conclusion

In this article, we answered some common questions related to the topic of factoring polynomial functions using the Remainder Theorem. We hope that this article has been helpful in clarifying any confusion you may have had about the topic.

Final Answer

The final answer is: $\boxed{(x-6)(x-10)(x+12)}$