Lynn Marecek
ISBN #9780998625713
1st Edition
6,645 Questions
Homework Questions
This chapter covers the foundational aspects of polynomials including identifying different types (monomials, binomials, and trinomials), determining degrees, and performing operations such as addition, subtraction, and evaluation. It also introduces the properties of exponents, which are essential when simplifying and manipulating polynomial expressions. These skills are not only central to algebra but also have practical applications in engineering, design, and various scientific fields.
1
Identify and classify polynomials, monomials, binomials, and trinomials.
2
Determine the degree of polynomials by analyzing the exponents of their terms.
3
Add and subtract monomials and polynomials by combining like terms.
4
Evaluate polynomials for given variable values.
5
Apply multiplication properties of exponents to simplify expressions.
CONCEPT
DEFINITION
Polynomial
An expression formed by the addition or subtraction of one or more monomials.
Monomial
A polynomial with exactly one term, typically in the form ax^m where a is a constant and m is a positive whole number.
Binomial
A polynomial with exactly two terms.
Trinomial
A polynomial with exactly three terms.
Degree
The highest sum of the exponents of the variables in any term of a polynomial; for a constant, the degree is 0.
Exponent
Indicates how many times the base is multiplied by itself; used in expressing the power of a number or variable.
Product Property of Exponents
If multiplying like bases, add the exponents: a^m * a^n = a^(m+n).
Power Property of Exponents
Raising a power to a power multiplies the exponents: (a^m)^n = a^(m*n).
Product to a Power Property
When raising a product to an exponent: (ab)^n = a^n * b^n.
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial. (a) $81 b^{5}-24 b^{3}+1$ (b) $5 c^{3}+11 c^{2}-c-8$ (c) $\frac{14}{15} y+\frac{1}{7}$ (a) 5 (e) $4 y+17$
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial. (a) $x^{2}-y^{2}$ (b) $-13 c^{4}$ (c) $x^{2}+5 x-7$ (d) $x^{2} y^{2}-2 x y+8$ (e) 19
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial. (a) $8-3 x$ (b) $z^{2}-5 z-6$ (c) $y^{3}-8 y^{2}+2 y-16$ (d) $81 b^{5}-24 b^{3}+1$ (e) -18
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial. a) $11 y^{2}$ (b) -73 (c) $6 x^{2}-3 x y+4 x-2 y+y^{2}$ (d) $4 y+17$ (e) $5 c^{3}+11 c^{2}-c-8$
In the following exercises, determine the degree of each polynomial. a) $6 a^{2}+12 a+14$ (b) $18 x y^{2} z$ (c) $5 x+2$ (d) $y^{3}-8 y^{2}+2 y-16$ (e) -24
QUESTION
Simplify (5y² - 3y + 15) + (3y² - 4y - 11).
STEP-BY-STEP ANSWER:
Step 1: Identify like terms. The y² terms: 5y² and 3y², the y terms: -3y and -4y, and the constants: 15 and -11. Step 2: Combine the like terms: 5y² + 3y² = 8y²; -3y - 4y = -7y; 15 - 11 = 4. Final Answer: 8y² - 7y + 4.
Adding Polynomials
Evaluate 5x² - 8x + 4 when x = 4.
Step 1: Substitute x = 4 into the polynomial: 5(4²) - 8(4) + 4. Step 2: Calculate the exponent: 4² = 16, then multiply: 5 * 16 = 80. Step 3: Multiply: 8 * 4 = 32. Step 4: Combine the results: 80 - 32 + 4 = 52. Final Answer: 52.
Evaluating a Polynomial
Find the degree of the polynomial 4x³ - 7x + 5.
Step 1: Identify the degrees of each term: 4x³ has degree 3, -7x has degree 1, and 5 is a constant with degree 0. Step 2: The highest degree among these is 3. Final Answer: The degree is 3.
Determining the Degree of a Polynomial