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Chapter 4: Problem 50

Find the value of each polynomial for \((a) x=2\) and \((b) x=-1\) \(x^{4}-6 x^{3}+x^{2}-x\)

Short Answer

Expert verified
For \(x = 2\), the value is -30. For \(x = -1\), the value is 9.

Step by step solution

01

Understand the polynomial

The given polynomial is \(x^{4} - 6x^{3} + x^{2} - x\). We need to find its value for two different values of \(x\): 2 and -1.
02

Substitute \(x = 2\)

Replace \(x\) with 2 in the polynomial: \((2)^{4} - 6(2)^{3} + (2)^{2} - (2)\).
03

Simplify the expression for \(x = 2\)

Calculate each term separately and then combine them: \[(2)^{4} = 16\]\[-6(2)^{3} = -6(8) = -48\]\[(2)^{2} = 4\]\[-(2) = -2\]Add them together: \[16 - 48 + 4 - 2 = -30\]
04

Result for \(x = 2\)

The value of the polynomial for \(x = 2\) is -30.
05

Substitute \(x = -1\)

Replace \(x\) with -1 in the polynomial: \((-1)^{4} - 6(-1)^{3} + (-1)^{2} - (-1)\).
06

Simplify the expression for \(x = -1\)

Calculate each term separately and then combine them: \[(-1)^{4} = 1\]\[-6(-1)^{3} = -6(-1) = 6\]\[(-1)^{2} = 1\]\[-(-1) = 1\]Add them together: \[1 + 6 + 1 + 1 = 9\]
07

Result for \(x = -1\)

The value of the polynomial for \(x = -1\) is 9.

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Key Concepts

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

Polynomial
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it's like a multivariable equation. Here we have a polynomial in the variable, denoted by the expression: \(x^{4} - 6x^{3} + x^{2} - x\). This polynomial has four terms:
  • \(x^{4}\)
  • \(-6x^{3}\)
  • \(x^{2}\)
  • \(-x\)
Each term involves an exponent of the variable \(x\) which tells us the degree of the polynomial. For instance, \(x^{4}\) means \(x\) is raised to the power of 4.
Substitution
Substitution simply means replacing the variable in the polynomial with a given value. To find the value of the polynomial \(x^{4} - 6x^{3} + x^{2} - x\) for \(x = 2\), we replace every instance of \(x\) with 2. The modified polynomial becomes: \((2)^{4} - 6(2)^{3} + (2)^{2} - (2)\). Similarly, for \(x = -1\), we substitute -1 for every \(x\) in the polynomial, resulting in \((-1)^{4} - 6(-1)^{3} + (-1)^{2} - (-1)\). This step is crucial as it sets up the polynomial for further calculations.
Simplification
Simplification involves calculating and combining all the terms in the polynomial. For \(x = 2\), we calculate:
  • \((2)^{4} = 16\)
  • \(-6(2)^{3} = -48\)
  • \((2)^{2} = 4\)
  • \(-2\)
Then, we combine the results: \(16 - 48 + 4 - 2\), which equals -30. For \(x = -1\), we have:
  • \((-1)^{4} = 1\)
  • \(-6(-1)^{3} = 6\)
  • \((-1)^{2} = 1\)
  • \(-(-1) = 1\)
Combining these, we get \(1 + 6 + 1 + 1 = 9\). Simplifying each term separately makes the problem manageable and reduces errors.
Exponents
Exponents are an important part of polynomials. An exponent tells us how many times a number (the base) is multiplied by itself. For example:
  • \((2)^{4}\) means \(2\times2\times2\times2 = 16\)
  • \((-1)^{4}\) means \(-1\times-1\times-1\times-1 = 1\)
  • \((2)^{3}\) means \(2\times2\times2 = 8\)
  • \((-1)^{2}\) means \(-1\times-1 = 1\)
Understanding exponents will help you simplify the polynomial more easily. Exponents also affect the degree of the polynomial, with higher exponents indicating a polynomial of higher degree.

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Most popular questions from this chapter

Multiply each of the following. (a) \((x-1)\left(x^{2}+x+1\right)\) (b) \((y-2)\left(y^{2}+2 y+4\right)\) Describe the final product as "the ___________of two_______."

If an object is projected upward under certain conditions, its height in feet is given by the trinomial $$-16 t^{2}+60 t+80$$ where \(t\) is in seconds. Evaluate this trinomial for \(t=2.5 .\) Use the result to fill in the blanks: If ________ seconds have elapsed, then the height of the object is ____________ feet.

Light travels at a speed of \(1.86 \times 10^{5} \mathrm{mi}\) per sec. When Venus is \(6.68 \times 10^{7} \mathrm{mi}\) from the sun, how long (in seconds) does it take light to travel from the sun to Venus? (Data from The World Almanac and Book of Facts.)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (x+7)^{2} $$

Graph each equation by completing the table of values. $$ \begin{aligned} &y=-x^{2}+4\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$
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