Problem 22

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(3 x^{3}-5 x^{2}+2 x+3\right) \div(x+2) $$

Step-by-Step Solution

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Answer

Quotient: $3x^2 - 11x + 24$, Remainder: $-45$.

1Step 1: Set Up for Synthetic Division

In synthetic division, write the coefficients of the dividend polynomial: $3x^3 - 5x^2 + 2x + 3$. So, we have 3, -5, 2, and 3. Also, note the divisor $x + 2$, and change $+2$ to $-2$ because we need the opposite sign of the constant term of our divisor.

2Step 2: Perform Synthetic Division

Write $-2$ on the left and draw a horizontal line. Bring down the first coefficient, which is 3. Multiply it by $-2$ and write the result under the next coefficient. This gives $-6$. Add $-5$ and $-6$ to get $-11$. Repeat this process: multiply $-11$ by $-2$ to get 22, add to 2 to get 24. Multiply 24 by $-2$ to get $-48$, and add to 3 to get $-45$.

3Step 3: Interpret the Result

The final row of numbers after doing synthetic division represents the coefficients of the quotient polynomial and the remainder. Here we have 3, -11, 24, and $-45$. Therefore, the quotient is $3x^2 - 11x + 24$ and the remainder is $-45$.

4Step 4: Write the Final Quotient and Remainder

The quotient of the division $(3x^3 - 5x^2 + 2x + 3) \div (x + 2)$ is $3x^2 - 11x + 24$, with a remainder of $-45$. Thus, the division statement is: \[(3x^3 - 5x^2 + 2x + 3) = (x + 2)(3x^2 - 11x + 24) - 45\]

Key Concepts

Polynomial DivisionQuotient and RemainderAlgebraic Expressions

Polynomial Division

Polynomial division is a process used to divide one polynomial by another, similar to how we perform division with regular numbers.
This method helps break down complex polynomials into simpler terms by finding a quotient and a remainder.
There are primarily two types of polynomial division: long division and synthetic division.

Long division: Works much like the long division method used in arithmetic, providing a detailed step-by-step approach.
Synthetic division: A shortcut method, used when the divisor is a linear polynomial, which is the case for our exercise where divisor is $x + 2$.

Both methods aim to express the original polynomial as the product of the divisor and the quotient plus the remainder. For the exercise at hand, using synthetic division simplifies the process immensely.

Quotient and Remainder

When dividing polynomials, just like dividing numbers, the result can be expressed in terms of a quotient and a remainder.
The quotient is the result of the division while the remainder is what is left over, similar to seeing how many times a number fits into another, with leftover.With synthetic division, the coefficients in the bottom row (except the last one) form the quotient.
The last number is the remainder.

Quotient: In our case, it is the polynomial $3x^2 - 11x + 24$.
Remainder: Here, it is the constant $-45$.

The equation including the remainder can thus be written as: \[ (3x^3 - 5x^2 + 2x + 3) = (x + 2)(3x^2 - 11x + 24) - 45 \] which clearly shows how the original polynomial is divided into the other components.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators (like plus and minus signs).
Polynomials are a specific type of algebraic expression that involves sums of powers of variables, each multiplied by a coefficient.In our current exercise, we worked with the polynomial expression $3x^3 - 5x^2 + 2x + 3$.
Breaking down this expression using division helps us see its structure by separating it into simpler parts, like quotient and remainder.Key components of handling algebraic expressions include:

Identifying coefficients: The numerical factors in front of each term (e.g., 3, -5, 2, and 3 in your polynomial).
Understanding variables: These represent unknown values and are typically letters like $x$.
Recognizing powers/exponents: They indicate how many times a variable is multiplied by itself (e.g., $x^3$, $x^2$).

Understanding these concepts lays the foundation for simplifying and solving algebraic problems reliably.

Problem 22

Other exercises in this chapter

Problem 22

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ x^{3}-3 x^{2}-10 x+24=0 $$

View solution

Problem 22

For the following exercises, determine the end behavior of the functions. $$ f(x)=3 x^{2}+x-2 $$

View solution

Problem 22

For the following exercises, find the $x$ - or $t$ -intercepts of the polynomial functions. $$ f(x)=x^{6}-3 x^{4}-4 x^{2} $$

View solution

Problem 22

Determine the end behavior of the functions. $$f(x)=3 x^{2}+x-2$$

View solution