Problem 1 Use synthetic substitution to ev... [FREE SOLUTION]

Chapter 3: Problem 1

Use synthetic substitution to evaluate $P(x)$ for the given values of $x$. $$P(x)=x^{4}+2 x^{3}-3 x^{2}-4 x-20, x=2, x=-3$$

Short Answer

Expert verified

Both $P(2)$ and $P(-3)$ are equal to -8.

Step by step solution

Synthetic substitution preparation for x = 2

To evaluate the polynomial $P(x) = x^4 + 2x^3 - 3x^2 - 4x - 20$ for $x = 2$, first list the coefficients of the polynomial: 1 (for $x^4$), 2 (for $x^3$), -3 (for $x^2$), -4 (for $x$), and -20 (the constant term). Drop the leading coefficient (1) to start the synthetic division process.

Perform synthetic substitution for x = 2

Write $2$ on the left and draw a vertical line separating it from the coefficients $1, 2, -3, -4, -20$. Carry down the 1. Multiply 1 by 2 to get 2, then add to the next coefficient: $2 + 2 = 4$. Continue: multiply 4 by 2 to get 8, add to -3 to get 5. Multiply 5 by 2 to get 10, add to -4 to get 6. Multiply 6 by 2 to get 12, add to -20 to get -8. The result is -8, meaning $P(2) = -8$.

Synthetic substitution preparation for x = -3

Now, evaluate $P(x)$ for $x = -3$. Use the same set of coefficients: 1, 2, -3, -4, -20. Begin by writing $-3$ on the left and repeat the synthetic substitution process.

Perform synthetic substitution for x = -3

Carry down the initial 1. Multiply 1 by -3 to get -3, add to the next coefficient: $2 + (-3) = -1$. Multiply -1 by -3 to get 3, add to -3 to get 0. Multiply 0 by -3 to get 0, add to -4 to get -4. Multiply -4 by -3 to get 12, add to -20 to get -8. The result is -8, showing that $P(-3) = -8$.

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Key Concepts

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

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial for a specific value of its variable. Consider the polynomial $ P(x) = x^4 + 2x^3 - 3x^2 - 4x - 20 $. To evaluate this polynomial at $ x = 2 $, you substitute $ x $ with 2 in the polynomial and calculate the result:

First, calculate $ 2^4 = 16 $.
Next, $ 2 \times 2^3 = 16 $.
Then, $ -3 \times 2^2 = -12 $.
And $ -4 \times 2 = -8 $.
Finally, add the constant $ -20 $.

Add these values to find that $ P(2) = -8 $. An alternative and quicker way to evaluate a polynomial is through synthetic substitution, which simplifies the arithmetic efficiently.

Synthetic Division

Synthetic division is a streamlined method for dividing polynomials and evaluating them. Unlike traditional long division, it requires less writing and is faster. This process is particularly useful when you need to evaluate a polynomial at a specific value, almost like a shortcut.Here are the basic steps:

Drop the leading coefficient to start the process.
Multiply this number by the value you are substituting and add it to the next coefficient.
Repeat this process through all coefficients.

For $ P(x) = x^4 + 2x^3 - 3x^2 - 4x - 20 $ and $ x = 2 $, start with 1 (the coefficient of $ x^4 $). Multiply and add sequentially along the coefficients until you reach the last number, which gives the evaluated result. Using this method, both $ P(2) $ and $ P(-3) $ result in $-8$. This consistency helps verify that each step in synthetic division is accurate.

Polynomial Coefficients

In a polynomial, coefficients are the numerical factors that multiply the variable terms. In the polynomial $ P(x) = x^4 + 2x^3 - 3x^2 - 4x - 20 $, the coefficients are critically important in the evaluation process.

The coefficient of $ x^4 $ is 1.
The coefficient of $ x^3 $ is 2.
The coefficient of $ x^2 $ is -3.
The coefficient of $ x $ is -4.
The constant term is -20.

When using synthetic substitution or division, these coefficients are systematically manipulated to efficiently compute the polynomial's value at certain points. Understanding how coefficients are used helps not only in simplifying calculations but also in ensuring that results are consistent and accurate. This highlights their fundamental role in both polynomial operations and solving.

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91影视

Use synthetic substitution to evaluate \(P(x)\) for the given values of \(x\). $$P(x)=x^{4}+2 x^{3}-3 x^{2}-4 x-20, x=2, x=-3$$

Short Answer

Step by step solution

Synthetic substitution preparation for x = 2

Perform synthetic substitution for x = 2

Synthetic substitution preparation for x = -3

Perform synthetic substitution for x = -3

Key Concepts

Polynomial Evaluation

Synthetic Division

Polynomial Coefficients

One App. One Place for Learning.

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