Problem 39 Use synthetic division and the R... [FREE SOLUTION]

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Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 39

Use synthetic division and the Remainder Theorem to evaluate $P(c)$. $$P(x)=4 x^{2}+12 x+5, \quad c=-1$$

Short Answer

Expert verified

The value of $P(-1)$ is -3.

Step by step solution

Setup the Synthetic Division

Write down the coefficients of the polynomial $P(x) = 4x^2 + 12x + 5$ in order: 4, 12, 5. The value of $c$ given is -1, which will be used in the synthetic division process below the division line.

Perform Synthetic Division

- Draw a horizontal line and place the coefficients 4, 12, and 5 in a row.- Write the value of $c = -1$ to the left.- Bring down the first coefficient (4) directly under the line.- Multiply this 4 by $c = -1$ and write the result under the second coefficient (12), then add them to get the new number below (which is 8).- Repeat the multiplication by $-1$ with the new number (8), place the result under the third coefficient (5), add them to get the final remainder, which is your answer (this case, -3).

Remainder Interpretation

In synthetic division, the last number you derived, in this case, -3, represents the remainder of the division. According to the Remainder Theorem, this remainder is equal to $P(-1)$. Hence, $P(-1) = -3$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Remainder Theorem

The Remainder Theorem is a handy tool when dealing with polynomials. It tells us that if a polynomial $ P(x) $ is divided by $ x-c $, then the remainder of this division is the same as evaluating the polynomial at $ c $. In simpler terms, if you plug $ c $ into the polynomial, the result is the remainder you'd get from division. For our exercise, after performing synthetic division on $ P(x) = 4x^2 + 12x + 5 $ using $ c = -1 $, we found the remainder to be -3. Therefore, by the Remainder Theorem, $ P(-1) = -3 $. This concept helps us quickly find specific values of a polynomial without needing to do full polynomial division.

Polynomial Evaluation

Evaluating a polynomial simply means substituting a given value for the variable in the polynomial and simplifying. For instance, if we want to determine $ P(-1) $ in the polynomial $ P(x) = 4x^2 + 12x + 5 $, we substitute $-1$ in place of $x$:

$ P(-1) = 4(-1)^2 + 12(-1) + 5 $
- First, calculate $(-1)^2$ which is 1.
- Then, proceed to find $ 4 \times 1 + 12 \times (-1) + 5 $ .
- This simplifies to $ 4 - 12 + 5 $.
Finally, $ P(-1) = -3 $, corroborating the result obtained by synthetic division and the Remainder Theorem. Using synthetic division streamlines this process by reducing multiple steps into a quicker algorithm.

Synthetic Division Steps

Synthetic division is a shortcut method for dividing polynomials that is easier and faster than long division under certain circumstances. Here are the steps to follow:

Identify the coefficients of your polynomial. For instance, in $ P(x)=4x^2+12x+5 $ the coefficients are 4, 12, and 5.
Write down these coefficients in a sequence.
Next to them, place the value of $ c $ from $ x-c $. Here, this is $-1$.
Start by bringing down the leading coefficient (4) to below the line.
Multiply this by $-1$ to get -4 and add it to the next coefficient (12), which gives you 8.
Repeat the multiplication with 8 and $-1$, to get -8. Add this to the next coefficient (5) to yield -3.

The final number is the remainder, confirming that $ P(-1) = -3 $ and is used as part of polynomial evaluation.

Polynomial Coefficients

Polynomial coefficients are the numerical factors of each term in a polynomial. These integers or real numbers form the backbone of polynomial expressions by accompanying each degree of the variable. In our polynomial $ P(x) = 4x^2 + 12x + 5 $, the coefficients are:

4: The coefficient of $x^2$
12: The coefficient of $x$
5: The constant term (also interpretable as the coefficient of $x^0$)

These coefficients are used in synthetic division to systematically evaluate the polynomial at a specific point. By understanding and correctly identifying these terms, you can correctly apply synthetic division and other polynomial operations with ease.

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91影视

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=4 x^{2}+12 x+5, \quad c=-1$$

Short Answer

Step by step solution

Setup the Synthetic Division

Perform Synthetic Division

Remainder Interpretation

Key Concepts

Remainder Theorem

Polynomial Evaluation

Synthetic Division Steps

Polynomial Coefficients

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