Final answer:
Each term in an algebraic expression has a coefficient, variable, and power. The polynomial function is built by combining such terms and its degree is the highest exponent of the variable across all terms. Polynomial functions are graphed to visualize their behavior and relationships among terms.
Step-by-step explanation:
In mathematics, particularly in algebra, an expression is made up of terms. Each term has a coefficient, a variable (unless it is a constant term), and an exponent indicating the power of the variable.
Identification of Terms, Coefficients, Variables, and Powers
1. For the expression y^4 + 13, we have two terms. The first term is y^4, which has the coefficient 1 (implied), the variable y, and the power 4. The second term is the constant 13, with no variable or power associated.
2. The expression 8c^3 − c^2 + 8c contains three terms. The first term 8c^3 has a coefficient of 8, variable c, and power 3. The second term, −c^2, has a coefficient of −1, variable c, and power 2. Lastly, the term 8c has a coefficient of 8, variable c, and power 1.
3. For 12z^5 + 9z^2 − z − 7, there are four terms. 12z^5 has a coefficient of 12, variable z, and power 5. 9z^2 has a coefficient of 9, variable z, and power 2. −z has a coefficient of −1 (implied), variable z, and power 1. The constant term − 7 has no variable or power.
4. In the expression −5m^10 + m^8 + 5m^6 − m^4, we observe four terms again. The first term −5m^10 includes a coefficient of −5, variable m, and power 10. m^8 has an implied coefficient of 1, variable m, and power 8. 5m^6 has a coefficient of 5, variable m, and power 6. The last term −m^4 has a coefficient of −1 (implied), variable m, and power 4.
Polynomial Functions and Their Degrees
5. The polynomial function would be 30x + x^8 − 3x^3, with the degree being 8 (since the highest power of the variable x is 8).
6. For the terms given, the function is 14x^2 − 6x^3 + 10 − x^6. The degree is 6, as this is the highest exponent among the terms.
7. The polynomial −x^5 + 2x + 22 + x^7 + 7x^3 is written with its terms in standard form, and the degree is 7 because x^7 is the term with the highest exponent.
In general, when dealing with polynomial functions, we arrange the terms in descending order of their powers and identify the highest exponent to determine the degree of the polynomial. This process helps in graphing polynomials and understanding their behavior.