1. Warm Up Evaluate. 1. –24 2. (–24) Simplify each expression.
3. x – 2(3x – 1)
4. 3(y2 + 6y)
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2. A monomial is a number or a product of numbers and variables with whole number exponents. A polynomial is a monomial or a sum or difference of monomials. Each monomial in a polynomial is a term. Because a monomial has only one term, it is the simplest type of polynomial.
Polynomials have no variables in denominators or exponents, no roots or absolute values of variables, and all variables have whole number exponents. 1 2 a7
Polynomials:
3x4
2z12 + 9z3
0.15x101
3t2 – t3 8
5y2 1 2
Not polynomials: 3x
|2b3 – 6b|
m0.75 – m
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3. The degree of a polynomial is the largest of the exponents of the variables.1 2 a7
Polynomials:
3x4
2z12 + 9z3
0.15x101
3t2 – t3
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4. Polynomial Functions
A polynomial function is a function whose rule is a polynomial.
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5. Example 1 f(x) = 3x3 – 18x + 45 Evaluate f(0) and f(200)
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6. Evaluate h(x) = 0.4x2 – 1.2x + 7.5 for x = 0 and x = 3.
Example 2
Evaluate h(x) = 0.4x2 – 1.2x for x = 0 and x = 3.
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7. Real Roots of a Polynomial Function
You can find the roots, or solutions, of the polynomial equation P (x) = 0 by setting each factor equal to 0 and solving for x
Roots are the x values, where the graph of the polynomial function crosses the x-axis.
Roots are also called real zeros.
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8. Example 3
Here is a graph each polynomial function on a calculator. Identify the number of real zeros. B.
A. f(x) = 2x3 – 3x
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9. Example 4
Here is a graph each polynomial function on a calculator. Identify the number of real zeros.
a. f(x) = 6x3 + x2 – 5x + 1
b. f(x) = 3x2 – 2x + 2
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10. Example 5
Here is a graph each polynomial function on a calculator. Identify the number of real zeros.
c. g(x) = x4 – 3
d. h(x) = 4x4 – 16x2 + 5
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