Select The Correct Answer.One Factor Of The Polynomial $3x^3 + 20x^2 - 21x + 88$ Is $(x + 8)$. What Is The Other Factor Of The Polynomial? (Note: Use Long Division)A. $(3x^2 - 4x + 11)$B. $(3x^2 -

Introduction

Polynomial equations are a fundamental concept in algebra, and solving them can be a challenging task. In this article, we will focus on one specific problem: given a polynomial equation and one of its factors, we need to find the other factor. We will use the long division method to solve this problem.

Understanding Polynomial Equations

A polynomial equation is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The general form of a polynomial equation is:

$$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

where $a_n \neq 0$ and $n$ is a non-negative integer.

Factors of Polynomial Equations

A factor of a polynomial equation is an expression that divides the polynomial without leaving a remainder. In other words, if we have a polynomial equation $p(x)$ and a factor $f(x)$, then $f(x)$ divides $p(x)$ if and only if $p(x) = f(x)q(x)$ for some polynomial $q(x)$.

Long Division Method

The long division method is a step-by-step process for dividing a polynomial by another polynomial. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, then multiplying the entire divisor by the result and subtracting it from the dividend. We repeat this process until we have a remainder that is of lower degree than the divisor.

Solving the Problem

We are given a polynomial equation $3x^3 + 20x^2 - 21x + 88$ and one of its factors $(x + 8)$. We need to find the other factor of the polynomial.

To solve this problem, we will use the long division method. We will divide the polynomial by the given factor and find the quotient.

Step 1: Divide the highest degree term of the dividend by the highest degree term of the divisor

The highest degree term of the dividend is $3x^3$ and the highest degree term of the divisor is $x$. Therefore, we divide $3x^3$ by $x$ to get $3x^2$.

Step 2: Multiply the entire divisor by the result and subtract it from the dividend

We multiply the entire divisor $(x + 8)$ by $3x^2$ to get $3x^3 + 24x^2$. We subtract this from the dividend to get:

$$20x^2 - 21x + 88 - (3x^3 + 24x^2)$$

Simplifying this expression, we get:

$$-4x^2 - 21x + 88$$

Step 3: Repeat the process

We repeat the process by dividing the highest degree term of the dividend by the highest degree term of the divisor. The highest degree term of the dividend is $-4x^2$ and the highest degree term of the divisor is $x$. Therefore, we divide $-4x^2$ by $x$ to get $-4x$.

We multiply the entire divisor $(x + 8)$ by $-4x$ to get $-4x^2 - 32x$. We subtract this from the dividend to get:

$$-21x + 88 - (-4x^2 - 32x)$$

Simplifying this expression, we get:

$$4x^2 + 13x + 88$$

Step 4: Repeat the process again

We repeat the process again by dividing the highest degree term of the dividend by the highest degree term of the divisor. The highest degree term of the dividend is $4x^2$ and the highest degree term of the divisor is $x$. Therefore, we divide $4x^2$ by $x$ to get $4x$.

We multiply the entire divisor $(x + 8)$ by $4x$ to get $4x^2 + 32x$. We subtract this from the dividend to get:

$$13x + 88 - (4x^2 + 32x)$$

Simplifying this expression, we get:

$$-19x + 88$$

Step 5: Repeat the process again

We repeat the process again by dividing the highest degree term of the dividend by the highest degree term of the divisor. The highest degree term of the dividend is $-19x$ and the highest degree term of the divisor is $x$. Therefore, we divide $-19x$ by $x$ to get $-19$.

We multiply the entire divisor $(x + 8)$ by $-19$ to get $-19x - 152$. We subtract this from the dividend to get:

$$88 - (-19x - 152)$$

Simplifying this expression, we get:

$$240 + 19x$$

Since the degree of the remainder is lower than the degree of the divisor, we stop the process.

Finding the Other Factor

We have found the quotient of the long division to be $3x^2 - 4x + 11$. Therefore, the other factor of the polynomial is $3x^2 - 4x + 11$.

The final answer is: $\boxed{3x^2 - 4x + 11}$

Introduction

Polynomial equations are a fundamental concept in algebra, and solving them can be a challenging task. In this article, we will focus on one specific problem: given a polynomial equation and one of its factors, we need to find the other factor. We will use the long division method to solve this problem.

Understanding Polynomial Equations

A polynomial equation is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The general form of a polynomial equation is:

$$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

where $a_n \neq 0$ and $n$ is a non-negative integer.

Factors of Polynomial Equations

A factor of a polynomial equation is an expression that divides the polynomial without leaving a remainder. In other words, if we have a polynomial equation $p(x)$ and a factor $f(x)$, then $f(x)$ divides $p(x)$ if and only if $p(x) = f(x)q(x)$ for some polynomial $q(x)$.

Long Division Method

The long division method is a step-by-step process for dividing a polynomial by another polynomial. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, then multiplying the entire divisor by the result and subtracting it from the dividend. We repeat this process until we have a remainder that is of lower degree than the divisor.

Solving the Problem

We are given a polynomial equation $3x^3 + 20x^2 - 21x + 88$ and one of its factors $(x + 8)$. We need to find the other factor of the polynomial.

To solve this problem, we will use the long division method. We will divide the polynomial by the given factor and find the quotient.

Step 1: Divide the highest degree term of the dividend by the highest degree term of the divisor

The highest degree term of the dividend is $3x^3$ and the highest degree term of the divisor is $x$. Therefore, we divide $3x^3$ by $x$ to get $3x^2$.

Step 2: Multiply the entire divisor by the result and subtract it from the dividend

We multiply the entire divisor $(x + 8)$ by $3x^2$ to get $3x^3 + 24x^2$. We subtract this from the dividend to get:

$$20x^2 - 21x + 88 - (3x^3 + 24x^2)$$

Simplifying this expression, we get:

$$-4x^2 - 21x + 88$$

Step 3: Repeat the process

We repeat the process by dividing the highest degree term of the dividend by the highest degree term of the divisor. The highest degree term of the dividend is $-4x^2$ and the highest degree term of the divisor is $x$. Therefore, we divide $-4x^2$ by $x$ to get $-4x$.

We multiply the entire divisor $(x + 8)$ by $-4x$ to get $-4x^2 - 32x$. We subtract this from the dividend to get:

$$-21x + 88 - (-4x^2 - 32x)$$

Simplifying this expression, we get:

$$4x^2 + 13x + 88$$

Step 4: Repeat the process again

We repeat the process again by dividing the highest degree term of the dividend by the highest degree term of the divisor. The highest degree term of the dividend is $4x^2$ and the highest degree term of the divisor is $x$. Therefore, we divide $4x^2$ by $x$ to get $4x$.

We multiply the entire divisor $(x + 8)$ by $4x$ to get $4x^2 + 32x$. We subtract this from the dividend to get:

$$13x + 88 - (4x^2 + 32x)$$

Simplifying this expression, we get:

$$-19x + 88$$

Step 5: Repeat the process again

We repeat the process again by dividing the highest degree term of the dividend by the highest degree term of the divisor. The highest degree term of the dividend is $-19x$ and the highest degree term of the divisor is $x$. Therefore, we divide $-19x$ by $x$ to get $-19$.

We multiply the entire divisor $(x + 8)$ by $-19$ to get $-19x - 152$. We subtract this from the dividend to get:

$$88 - (-19x - 152)$$

Simplifying this expression, we get:

$$240 + 19x$$

Since the degree of the remainder is lower than the degree of the divisor, we stop the process.

Finding the Other Factor

We have found the quotient of the long division to be $3x^2 - 4x + 11$. Therefore, the other factor of the polynomial is $3x^2 - 4x + 11$.

The final answer is: $\boxed{3x^2 - 4x + 11}$

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Q&A

Q: What is a polynomial equation?

A: A polynomial equation is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What is a factor of a polynomial equation?

A: A factor of a polynomial equation is an expression that divides the polynomial without leaving a remainder.

Q: How do you find the other factor of a polynomial equation using the long division method?

A: To find the other factor of a polynomial equation using the long division method, you divide the polynomial by the given factor and find the quotient.

Q: What is the long division method?

A: The long division method is a step-by-step process for dividing a polynomial by another polynomial. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, then multiplying the entire divisor by the result and subtracting it from the dividend.

Q: How do you divide a polynomial by another polynomial using the long division method?

A: To divide a polynomial by another polynomial using the long division method, you divide the highest degree term of the dividend by the highest degree term of the divisor, then multiply the entire divisor by the result and subtract it from the dividend.

Q: What is the quotient of the long division method?

A: The quotient of the long division method is the result of dividing the polynomial by the given factor.

Q: What is the remainder of the long division method?

A: The remainder of the long division method is the result of subtracting the product of the divisor and the quotient from the dividend.

Q: How do you find the other factor of a polynomial equation using the long division method?

A: To find the other factor of a polynomial equation using the long division method, you divide the polynomial by the given factor and find the quotient.

Q: What is the final answer to the problem?

A: The final answer to the problem is $3x^2 - 4x + 11$.

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Conclusion

In this article, we have discussed how to solve polynomial equations using the long division method. We have also provided a step-by-step guide on how to find the other factor of a polynomial equation using the long division method. We have also answered some common questions related to polynomial equations and the long division method.