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Chapter 5: Problem 58

Evaluate polynomial for \(x=3\) and for \(x=-3\). \(4 x^{2}-6 x+9\)

Short Answer

Expert verified
For x=3, the value is 27. For x=-3, the value is 63.

Step by step solution

01

Identify the polynomial function

The given polynomial function is
02

Substitute x = 3 into the polynomial

Replace every occurrence of x with 3. The polynomial becomes: f(3) = 4(3)^2 - 6(3) + 9 Evaluate this expression.
03

Calculate the value for x = 3

First, calculate the squared term: 4(3)^2 = 4(9) = 36 Next, multiply: -6(3) = -18 Now, add and subtract the constants: 36 - 18 + 9 = 27
04

Substitute x = -3 into the polynomial

Replace every occurrence of x with -3. The polynomial becomes: f(-3) = 4(-3)^2 - 6(-3) + 9 Evaluate this expression.
05

Calculate the value for x = -3

First, calculate the squared term: 4(-3)^2 = 4(9) = 36 Next, multiply: -6(-3) = 18 Now, add the numbers together: 36 + 18 + 9 = 63

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Key Concepts

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

substitution method
The substitution method is key for evaluating polynomial functions. It involves replacing a variable with a given number. This number is then used to compute the value of the polynomial. For instance, if you have the polynomial function \(f(x) = 4x^2 - 6x + 9\), and you are asked to find the value at \(x = 3\), you would substitute every \(x\) in the polynomial with 3. Hence, \(f(3) = 4(3)^2 - 6(3) + 9\). This step helps in simplifying and solving the expression systematically.
polynomial functions
Polynomial functions are algebraic expressions composed of variables and coefficients. They are written in the form \(anx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\). Each term has a coefficient and the variable is raised to a power called the exponent. The highest power in the polynomial is the degree. In our example, \(4x^2 - 6x + 9\) is a polynomial of degree 2. Polynomial functions can be classified based on their degree, such as linear (degree 1), quadratic (degree 2), and cubic (degree 3). Evaluating them involves substituting values for the variables and performing arithmetic operations.
evaluating algebraic expressions
Evaluating algebraic expressions like polynomials involves substituting variables with numerical values and performing arithmetic to find a result. This process starts with identifying the given polynomial and the specific values for variables. For instance, to evaluate \(4x^2 - 6x + 9\) when \(x = 3\), replace \(x\) with 3. Calculate the result by following the order of operations (PEMDAS):

  • First, compute exponents: \(4(3)^2 = 4(9) = 36\).
  • Then, multiply coefficients by the substituted value: \(-6(3) = -18\).
  • Finally, add and subtract terms from left to right to get the result: \(36 - 18 + 9 = 27\).

Follow the same steps when substituting \(x = -3\) to get another value.

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