Math II: Types of Polynomials

1) 3) x² + y²

The given expression, x2 + y2, is a binomial because it consists of two terms, x2 and y2, which are separated by a plus sign. A binomial is a polynomial with two terms, and in this case, both terms are raised to the power of 2.

Explanation

The given expression, x2 + y2, is a binomial because it consists of two terms, x2 and y2, which are separated by a plus sign. A binomial is a polynomial with two terms, and in this case, both terms are raised to the power of 2.

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2) 4) 4xyz

The given expression, 4xyz, is a monomial. A monomial is an algebraic expression that consists of only one term. In this case, the term is 4xyz, which is a product of the constants 4, and the variables x, y, and z. Since there is only one term present, it qualifies as a monomial.

Explanation

The given expression, 4xyz, is a monomial. A monomial is an algebraic expression that consists of only one term. In this case, the term is 4xyz, which is a product of the constants 4, and the variables x, y, and z. Since there is only one term present, it qualifies as a monomial.

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3) 8) 3a + 2b

The expression 3a + 2b is a binomial because it consists of two terms, 3a and 2b, that are separated by a plus sign. In a binomial, there are only two terms, unlike a polynomial which can have more than two terms.

Explanation

The expression 3a + 2b is a binomial because it consists of two terms, 3a and 2b, that are separated by a plus sign. In a binomial, there are only two terms, unlike a polynomial which can have more than two terms.

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4) 9) 7a + b - 2

A trinomial is an algebraic expression with three terms. In the given expression, 7a + b - 2, we have three terms: 7a, b, and -2. Therefore, the expression can be classified as a trinomial.

Explanation

A trinomial is an algebraic expression with three terms. In the given expression, 7a + b - 2, we have three terms: 7a, b, and -2. Therefore, the expression can be classified as a trinomial.

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5) 10) 2m²n - mn²

The given expression 2m2n - mn2 is a binomial because it consists of two terms, 2m2n and mn2, connected by a subtraction operation. A binomial is an algebraic expression with two unlike terms separated by either addition or subtraction. In this case, the terms have different variables (m and n) and different exponents (2 and 1), making it a binomial.

Explanation

The given expression 2m2n - mn2 is a binomial because it consists of two terms, 2m2n and mn2, connected by a subtraction operation. A binomial is an algebraic expression with two unlike terms separated by either addition or subtraction. In this case, the terms have different variables (m and n) and different exponents (2 and 1), making it a binomial.

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6) 2) x² + 10x + 5

The given expression, x2 + 10x + 5, is a trinomial because it consists of three terms. In this case, the terms are x2, 10x, and 5. A trinomial is a polynomial with three terms, and it is classified as such based on the number of terms it contains.

Explanation

The given expression, x2 + 10x + 5, is a trinomial because it consists of three terms. In this case, the terms are x2, 10x, and 5. A trinomial is a polynomial with three terms, and it is classified as such based on the number of terms it contains.

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7) I. Identify each of the following expressions as monomial, binomial, trinomial, or polynomial (more than three terms).

1) 6a²b²c²

The given expression, 6a2b2c2, is a monomial. A monomial is an algebraic expression with only one term. In this case, the expression has only one term, which is 6a2b2c2. Therefore, it is classified as a monomial.

Explanation

The given expression, 6a2b2c2, is a monomial. A monomial is an algebraic expression with only one term. In this case, the expression has only one term, which is 6a2b2c2. Therefore, it is classified as a monomial.

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8) 7) am² + bm³ + cm³ + d

The given expression, am2 + bm3 + cm3 + d, represents a polynomial. A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. In this case, the expression contains variables (a, b, c) raised to positive integer exponents (2, 3) and a constant term (d). Hence, it satisfies the definition of a polynomial.

Explanation

The given expression, am2 + bm3 + cm3 + d, represents a polynomial. A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. In this case, the expression contains variables (a, b, c) raised to positive integer exponents (2, 3) and a constant term (d). Hence, it satisfies the definition of a polynomial.

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9) 5) am³ - (a-m)/3 + 17m

The given expression, am3 - (a-m)/3 + 17m, is a trinomial because it consists of three terms: am3, -(a-m)/3, and 17m. In a trinomial, there are three unlike terms separated by addition or subtraction. Here, the terms have different variables (a and m) and different exponents (3 and 1). Therefore, the expression can be classified as a trinomial.

Explanation

The given expression, am3 - (a-m)/3 + 17m, is a trinomial because it consists of three terms: am3, -(a-m)/3, and 17m. In a trinomial, there are three unlike terms separated by addition or subtraction. Here, the terms have different variables (a and m) and different exponents (3 and 1). Therefore, the expression can be classified as a trinomial.

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10) 6) ab (m² - n³)

The given expression "ab (m2 - n3)" is a binomial because it consists of two terms, "ab" and "(m2 - n3)". A binomial is an algebraic expression with exactly two unlike terms connected by addition or subtraction. In this case, the terms "ab" and "(m2 - n3)" are connected by multiplication.

Explanation

The given expression "ab (m2 - n3)" is a binomial because it consists of two terms, "ab" and "(m2 - n3)". A binomial is an algebraic expression with exactly two unlike terms connected by addition or subtraction. In this case, the terms "ab" and "(m2 - n3)" are connected by multiplication.

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Math II: Types Of Polynomials